OPEN BANKING: CREDIT MARKET COMPETITION WHEN BORROWERS OWN THE DATA

By

Zhiguo He, Jing Huang, and Jidong Zhou

November 2020

COWLES FOUNDATION DISCUSSION PAPER NO. 2262

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

Box 208281 New Haven, Connecticut 06520-8281

http://cowles.yale.edu/

Open Banking: Credit Market Competition WhenBorrowers Own the Data∗

Zhiguo He† Jing Huang‡ Jidong Zhou§

November 13, 2020

Abstract

Open banking facilitates data sharing consented by customers who generate the data, with

a regulatory goal of promoting competition between traditional banks and challenger fintech

entrants. We study lending market competition when sharing banks’ customer data enables

better borrower screening or targeting by fintech lenders. Open banking could make the entire

financial industry better off yet leave all borrowers worse off, even if borrowers could choose

whether to share their data. We highlight the importance of equilibrium credit quality inference

from borrowers’ endogenous sign-up decisions. When data sharing triggers privacy concerns by

facilitating exploitative targeted loans, the equilibrium sign-up population can grow with the

degree of privacy concerns.

Keywords: Open banking, Data sharing, Banking competition, Digital economy, Winner’scurse, Privacy, Precision marketing

∗Preliminary and comments welcome; references incomplete. We thank the seminar participants at Hong KongShue Yan University. Zhiguo He acknowledges the financial support from the Center for Research in Security Pricesat the University of Chicago Booth School of Business. All errors are ours.†University of Chicago, Booth School of Business, and NBER, 5807 South Woodlawn Ave, Chicago, IL 60637.

E-mail: zhiguo.he@chicagobooth.edu.‡University of Chicago, Booth School of Business, 5807 South Woodlawn Ave, Chicago, IL 60637. E-mail:

jing.huang@chicagobooth.edu.§Yale University, School of Management, 165 Whitney Avenue, New Haven, CT 06511. E-mail: ji-

dong.zhou@yale.edu.

1 Introduction

The world is racing toward an era of open-data economy, thanks to the rapidly evolving informationand digital technology. Customer data—instead of being zealously kept within individual organi-zations or institutions in an isolated fashion—have become more “open” to external third parties,whenever customers who generate these data consent to share them. Open banking, an initiativeled by several governments, including Australia and the United Kingdom, leads such a shift towardthe open-data economy. As the global discussion unfolds, many practitioners and policy makersexpect “open banking” to represent perhaps the most transformative trend in the banking industryin the coming decade.

Although open banking can be viewed as part of broad endeavor by the European Union onconsumer privacy protection (typified by the General Data Protection Regulation), the core princi-ple of open banking does not stop at customer ownership of their own data, but more importantly,envision enabling customers to voluntarily share their financial data with other entities, via, say,application programming interfaces, or APIs (Second Payment Services Directive, PSD2).1 WhenDeloitte Insight conducted a survey on open banking in April 2019, it employed the following“descriptive” definition of open banking, which vividly captures its essence:2

Imagine you want to use a financial product offered by an organization other than yourbank. This product could be anything you feel would help you, such as an app that givesyou a full picture of your financial status, including expenses, savings, and investmentsor it could be a mortgage or line of credit. But for this product to be fully useful to you,it needs information from your bank, such as the amount of money you have comingin and going out of your accounts, how many accounts you have, how you spend yourmoney, how much interest you have earned or paid, etc. You then instruct your bankto share this information with this other institution or app. Should you wish to stopusing this product, you can instruct your bank to stop sharing your data at any givenpoint in time, with no strings attached. This concept is called open banking.

The idea to let borrowers decide if they want to share data with some third parties—especially com-peting fintech lenders may have profound implications on credit market competition and welfare.To the best of our knowledge, our paper is the first to study this question theoretically. Althoughthe role of information technology has been studied in the literature of credit market competition as

1The PSD2 in European Union mandates European banks create best practices in APIs, vendor integration,and data management. Loosely speaking, application programming interfaces, allow users to synchronize, link, andconnect databases; in the context of a banking system, they link a bank’s database (its customers’ information) withdifferent applications or programs, thus forming a network encouraging the promotion of services, payments, andproducts appropriate to each person.

2See endnote 1 on page 17 in Srinivas, Schoeps, and Jain (2019). The article is available at https://bit.ly/3mIdm2N. Open banking regulations that require banks to share their data with third parties upon customers’ consenthave been implemented in Europe, Australia, and many Asian countries such as Singapore and Japan. Though nosuch regulation exists in the U.S. today, major banks are already implementing open data sharing through APIs whichallow third parties to access to their data. Notable examples include JP Morgan Chase (https://bit.ly/3kMjB4V)and U.S. Bank (https://bit.ly/3oC4KfI).

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we will discuss later, our paper emphasizes that, under open banking, it is borrowers—rather thanlenders—who control lenders’ access to borrower information via their own data sharing decisions.This conceptual difference is the cornerstone of our analysis, and begets many interesting questionsregarding the welfare implications of open banking.

We consider a traditional bank and a fintech lender in competition with each other. Theyconduct independent but imperfect creditworthiness tests before making loan offers to borrowers.Each borrower can have a high or low credit quality, and the test yields a binary signal of their creditquality. This framework is based on Broecker (1990) and has been widely used to study lendingmarket competition as we will discuss later. Similar to common-value auctions, an importantfeature of this market is a winner’s curse (i.e., winning a borrower implies the possibility that therival lender has observed an unfavorable signal of the borrower’s credit quality). This winner’scurse essentially determines the lending cost. In equilibrium, the lender that has a better screeningability and so faces a less severe winner’s curse earns a positive profit in expectation, while theother one with a weaker screening ability earns a zero profit and sometimes declines to extend anoffer to a borrower even upon seeing a favorable signal.

This baseline credit market competition model is presented in Section 2. We use it to studythe impact of open banking on market performance. Traditional banks enjoy a great advantagefrom a vast amount of customer data they possess (say, from transaction accounts, direct depositactivities, etc). Fintech lenders are often equipped with limited data (usually restricted to socialactivities and profiles), but much more advanced data analysis algorithms; without enough data,however, a better algorithm does not yield more useful information. Therefore, in our benchmarkcase with no open banking, we assume that the bank has a better screening ability than the fintechlender. (We define screening ability as the joint outcome of data availability and data analysistechniques.) Open banking, by allowing borrowers to share their bank data, can greatly enhancethe competitiveness of the fintech lender as a “challenger.”

We study two types of data that borrowers can share via open banking. The first containsinformation on borrowers’ credit quality, which affects the lending cost of financial institutions.The other type of data potentially reveals borrowers’ preferences for the fintech loan, and these“privacy” data might enable the fintech to offer targeted loans to exploit borrowers.

Section 3 examines credit-quality data sharing. Once the fintech has an access to the bank’sdata, we assume that its screening ability is improved. Because the fintech has a more advanceddata analysis algorithm, it could even surpass the bank in screening borrowers, especially when italso has some independent data sources.3 The improvement of the fintech’s screening ability hastwo effects: First, as the fintech now can better identify a borrower’s true type, it helps high creditquality borrowers but hurts low credit quality borrowers. This is a standard “information effect.”Second, it also affects the extent of winner’s curse that each lender faces, and so the degree of lendingcompetition. This “strategic effect” can go either direction: lending competition will be intensified

3For example, Berg, Burg, Gombović, and Puri (2020) provide evidence that fintech lenders use a different sourceof information, digital footprints, to assess customers’ creditworthiness; digital footprints improve the predictivepower of traditional credit bureau data when combined with the latter.

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(softened) if the screening ability gap between the two lenders shrinks (expands). In particular, ifopen banking expands the screening ability gap sufficiently (i.e., if open banking “overempowers”the fintech), it will hurt both types of borrowers but improve industry profit. Reflecting on thecelebrated selling point that open banking promotes competition and benefits borrowers, we hencehighlight that data sharing may backfire and increase the competitiveness of the challenger lendertoo much.

The key question is then: can the very nature of open banking—borrowers deciding whetherto opt in to share their banking data with the fintech lender—prevent this perverse effect of openbanking on borrowers from happening? After all, borrowers will not act against their own interest.Our analysis with voluntary sign-up decisions provides a negative answer to this question. Weshow that provided that there exist some borrowers who never sign up due to prohibitively highsign-up costs (e.g., because they do not know how to use the new technology or have strongprivacy concerns), generically there exists a non-empty set of parameters under which in the only(nontrivial) equilibrium, proportionally more high-type borrowers sign up and all borrowers areleft strictly worse off compared to the regime of before open banking. Those who sign up sufferdue to weakened competition as a result of the enlarged lender asymmetry caused by data sharing;those who do not suffer due to an adverse equilibrium inference that opting-out signals poor creditquality.

Section 4 studies data sharing on customer preference privacy. We assume that borrowers aresubject to some random shocks under which they can only take the fintech’s loan. For instance,they happen to need a quick loan and only the fintech can process with “immediacy”; or they havereached the bank’s borrowing limit and so can only resort to the fintech. Open banking, with theaid of big data technology that can integrate, say, borrowers’ social data and digital footprintstogether with their bank account information, allows the fintech to identify these “privacy events.”Accordingly, the fintech can better take advantage of borrowers by performing precision marketing,or in other words, “delivering the right offer at the right time to the right customer.”4 To highlightthe effect of privacy events, we assume that open banking in this case does not enhance the fintech’sability to screen the credit quality of borrowers. The borrowers in privacy events resemble the“captured” consumers in Varian (1980) who consider only one seller’s offer.

When the probability of privacy events is sufficiently small, the small pool of captured borrowersis not enough to compensate the loss from the winner’s curse for the fintech lender, which has aninferior screening ability. Similar to the baseline model, the fintech hence earns a zero profit inequilibrium and sometimes does not make offers to borrowers with a good signal. Its profit fromcaptured borrowers is offset by the bank’s more aggressive bidding. (This is different from Varian(1980) where any firm with some captured consumers must earn a positive profit.) Open banking

4Transaction records from the borrower’s bank allow fintech lenders to infer the borrower’s more detailed demo-graphic and credit information, by analyzing, say, the income and occupation revealed from direct deposits, consump-tion habit, and other information. This inference, combined with browsing and location data and their much shorterloan application processing time, allows fintech lenders to assess and meet the borrower’s demand of “immediacy”—e.g., borrowers traveling abroad need loans in foreign currencies on the spot, or consumers on e-commerce platformswith impulse purchase needs.

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allows the fintech to identify borrowers in privacy events and charge them a monopoly interest rate.If there were no credit quality inference from sign-up decisions, high-type borrowers would opt outto avoid paying a predatory interest rate in privacy events, while the opposite holds for low-typeborrowers who repay much less often and hence care little about the interest rate. Due to thisstigma effect of associating signing up with low credit quality, nobody signs up in equilibrium afteropen banking.

On the contrary, when the probability of privacy events is sufficiently large, the results are ratherdifferent. Before open banking, thanks to a large pool of potential captured borrowers, the fintechnow earns a positive profit and always makes an offer upon a good signal. This is particularlyattractive to the low-type who only care about the chance of getting a loan—in fact, they neversign up for open banking to reveal privacy events. This leads to a favorable credit quality inferencefor signing-up, leading all high-type borrowers to sign up in equilibrium. Piecing this together withthe previous case, we predict a rising sign-up population or widespread open-banking adoptionwhen the probability of privacy events increases. This perhaps counterintuitive result is driven bythe signaling effect of sign-up decisions.

In the case of data sharing on preference based privacy, in terms of the impact on borrowerwelfare, we have a similar result as in the case of data sharing on credit quality information.That is, it is possible that all borrowers suffer from open banking in equilibrium. This happensfor an intermediate probability of privacy events, and again this is because sign-up decisions areintertwined with credit quality inference in equilibrium. Loosely speaking, those who sign up sufferdue to being exploited in privacy events;5 those who do not suffer due to an unfavorable creditquality inference.

Related Literature

Lending market competition with asymmetric information. Our paper is built on Broecker (1990),which studies lending market competition with screening tests. In Broecker (1990), banks are sym-metric and possess the same screening ability, while both our paper and Hauswald and Marquez(2003) consider asymmetric screening abilities.6 Hauswald and Marquez (2003) study the com-petition between an inside bank who can conduct credit screening and an outside bank who hasno access to screening. They consider the possibility of information spillover to the outside bank,which reduces the inside bank’s information advantage and benefits borrowers. When open bank-ing facilitates sharing data on customer credit quality, it has some connection to the information

5Compared to opting out, signing up for open banking makes high-type borrowers to be exploited by a monopolyinterest rate in the privacy event, whereas low-type borrowers are more likely to lose the chance of receiving a loanoffer in the non-privacy event.

6Lending market competition with asymmetric screening abilities is related to common-value auctions with asym-metrically informed bidders. The early papers include Milgrom andWeber (1982) and Engelbrecht-Wiggans, Milgrom,and Weber (1983); later papers such as Hausch (1987), Kagel and Levin (1999), and Banerjee (2005) explore informa-tion structures that allow each bidder to have some private information (which is the information structure adopted inBroecker (1990) and our paper). The common-valuation auction literature suggests that reducing the more-informedbidder’s information advantage tends to intensify competition and improve the seller’s revenue, a result emerging inour baseline model as well.

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spillover effect studied in Hauswald and Marquez (2003).Our paper differs from Hauswald and Marquez (2003) in three important aspects: First, in

our model, open banking can empower the fintech, the initial weak lender, so that it exceeds thetraditional bank in screening ability, which can harm all borrowers. Second, an important featureof open banking, which we highlight in this paper, is that customers have the control of whetherto share their data, and their sign-up decision itself can potentially reveal further creditworthinessinformation. Third, open banking can also reveal non-credit privacy information to the fintechlender. How this enables the fintech to make more targeted loan offers and affects lending marketcompetition has not been investigated in the literature.

Asymmetric credit market competition can also arise from the bank-firm relationship, as abank knows its existing customers better than a new competitor; this idea was explored by Sharpe(1990).7 In our model, information asymmetry before open banking exists for the same reason:traditional banks own the customer data that fintech lenders have no access to, so that even iffintech lenders have a better data processing algorithm, they screen borrowers less accurately.

Our paper is also related to the literature on credit information sharing among banks; e.g.,Pagano and Jappelli (1993) and Bouckaert and Degryse (2006).8 More broadly, lending marketcompetition with asymmetric information is important for studying many issues such as capitalrequirement (e.g., Thakor, 1996), borrowers’ incentives to improve project quality (e.g., Rajan,1992), information dispersion and relationship building (e.g., Marquez, 2002), credit allocation(e.g., Dell’Ariccia and Marquez, 2004, 2006), etc.

Fintechs. Our paper connects to the growing literature on fintech disruption (see, for instance,Vives, 2019, for a review of digital disruption in banking), in particular on fintech companiescompeting with traditional banks in originating loans.9 Berg, Burg, Gombović, and Puri (2020) findthat even simple digital footprints are informative in predicting consumer default, as a complementsource of information to traditional credit bureau scores. On studies that support the notion ofa competition relationship between fintech and bank in our paper, Fuster, Plosser, Schnabl, andVickery (2019) examine the mortgage market and provide evidence that fintech lenders’ technologyadvantage increases origination efficiency: the automated fintech lending system results in fasterprocessing and more elastic response to changes in borrower demand. Di Maggio and Yao (2020)

7In the two-period model analyzed in Sharpe (1990), asymmetric competition arises in the second period. Thecorrected analysis of the second-period competition with a mixed-strategy equilibrium is offered in Von Thadden(2004).

8These two papers differ from ours in terms of focus as well as framework. Pagano and Jappelli (1993) study acollective decision on information sharing among banks (e.g., by setting up a credit bureau) where each bank acts asa monopolist in a local market. A bank can tell its residential borrowers’ types and so offers type-dependent deals,but it does not know the types of borrowers who immigrate from other markets and so has to offer them a uniforminterest rate. Once customer information is shared, each bank can discriminate over different types of immigrantborrowers as well. Bouckaert and Degryse (2006) study banks’ individual incentives to share customer information.They argue that an incumbent bank has a strategic incentive to share partial customer information to reduce theentry of new competitors. In our paper, the sharing of bank customer data to the fintech is faciliated by open bankingregulation and importantly is controlled by customers themselves.

9Blockchain and its underlying distributed ledger technology are another important disruption force in today’sfinancial industry that has received great attention since the launch of Bitcoin. For related work on this topic, seeHe and Cong (2019); Biais, BisiÚre, Bouvard, and Casamatta (2019) and Abadi and Brunnermeier (2019).

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find that fintech lenders serve borrowers of decent credit quality by financing higher consumptionexpenditures, who then default ex post more frequently than similar borrowers with non-fintechlenders. Their paper suggests a story in which some borrowers’ desire for immediate consumptionwith fintech loans exacerbates their self-control issues to overborrow, a point that is consistentwith one interpretation of the captured customers that we study in Section 4.10 In related theorywork, Parlour, Rajan, and Zhu (2020) study a bank that operates in both payment service andcredit (loan) markets; the bank is a monopoly lender in the loan market, but competes withstand-alone fintechs on payment service. Parlour, Rajan, and Zhu (2020) stress that customers’payment services provide information about their credit quality. In their paper, customers knowthat switching their payment service to fintechs has consequences on their credit service (akin toborrowers’ data-sharing decisions in our model), but there is no equilibrium credit quality inference,which plays the key role in our analysis.11

Consumer privacy. Our paper also contributes to the burgeoning literature on consumer privacy(see, for instance, Acquisti, Taylor, and Wagman, 2016; Bergemann and Bonatti, 2019, for re-cent surveys), and is particularly related to work on the impact of letting consumers control theirown data. Recent research suggests that the market equilibrium consequence of consumer privacychoices is highly context dependent. For example, in a general equilibrium setup Jones and Tonetti(2020) argue that consumer data ownership often leads to broader data usage than in the case offirm ownership, improving welfare thanks to the non-rivalry of data use. Ichihashi (2020) considersa multi-product monopoly problem where consumers choose whether to share data about theirpreferences, which can be used by the seller for both product recommendations and price discrimi-nation.12 Liu, Sockin, and Xiong (2020) examine the implications of consumer privacy when thereis both a normal consumption good and a temptation good; data sharing shapes sellers’ marketingschemes for reaching target consumers, which improve the efficiency of the normal good but alsoinduce some behaviorally biased consumers to overconsume the temptation good.13

We contribute to this literature by studying consumer privacy choices in the lending market.With the interaction between credit quality information and preference information, our paper

10Buchak, Matvos, Piskorski, and Seru (2018) study the mortgage market and argue that banking regulation andtechnology advancement contributed to the significant growth of fintech lenders. Tang (2019) uses a regulatory changethat contracts bank credit as an exogenous shock and finds that P2P platforms substitutes with bank in the consumercredit market.

11Whether a consumer switches the payment service to fintechs is driven by her bank-affinity preference, which isassumed to be independent of her credit quality type.

12Ichihashi (2020) shows that the seller has an incentive to commit to not price discriminate to encourage consumerinformation sharing, but this can harm consumers in equilibrium because the firm may set a higher uniform priceanticipating better a match between consumers and products. Ali, Lewis, and Vasserman (2020) study a relatedproblem but in a different setup with single-product firms and argue that sharing preference information with firmscan benefit consumers by amplifying price competition.

13Liu, Sockin, and Xiong (2020) emphasizes the difference between two consumer privacy regulations, namelyGDPR in EU (opt-out as the default choice) and CCPA in California (opt-in as the default choice). Our study ofprivacy and targeted loans in Section 4 is also related to the literature on oligopolistic price discrimination (see,for instance, the survey by Stole, 2007). In our model only the fintech lender can make targeted offers, so thereis asymmetric oligopolistic price discrimination. Two major differences relative to this IO literature are: our creditmarket competition features the winner’s curse; in our model, consumers choose whether to disclose their preferenceinformation, rather than firms deciding whether to acquire consumer information.

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highlights the equilibrium inference from consumers’ sign-up decisions in open banking. Using datafrom an online travel intermediary, Aridor, Che, and Salz (2020) offer evidence that this type ofinference is well founded. They show that letting privacy-conscious consumers opt out of datasharing under GDPR increases the average value of the remaining consumers to advertisers.

2 The Baseline Model

This section introduces the basic model of credit market competition that will be used as a buildingblock in later sections when we study open banking. Table 1 in Appendix A.1 provides a detailedlist of the notation used in this paper. All proofs are found in the Appendix.

2.1 Borrowers

There is a continuum of risk-neutral borrowers of measure one. Each is looking for a loan that isnormalized to be 1. Borrowers differ in their default risk: a fraction θ ∈ (0, 1) of them are high-type(h) borrowers who will repay the loan with probability µh ∈ (0, 1], and the rest 1 − θ of them arelow-type (l) borrowers who will repay the loan with probability µl < µh. Each borrower’s type isthat borrower’s private information, but the type distribution is publicly known. Let

τ ≡ θ

1− θ

be the likelihood ratio of high-type over low-type borrowers in the population, which represents theaverage credit quality in the market.

We assume that the interest rate in the market never exceeds r. There are at least two interpre-tations of this assumption. Borrowers can be small business firms, each having a project in whichto invest but differing in the probability that their project will succeed. When the project succeeds,it yields a net return r, which is observable and contractible; when it fails, it yields nothing. As aresult, borrowers will never accept a loan with an interest rate above r. Alternatively, borrowerscan be ordinary consumers who need a loan to purchase a product but differ in the probabilitythat they will be able to repay the loan. (For instance, a consumer will default if she becomesunemployed, and consumers face different unemployment risks.) In this case we assume that theutility from consuming the product is sufficiently high for each type of consumer,14 but the interestrate is capped at r either due to interest rate regulation (e.g., usury laws) that prohibits excessivelyhigh rates of interest,15 or because of some exogenous outside options.

14In this case, for a borrower of type i ∈ {h, l}, denote by ui the utility from consuming the product. We assumethat δi ≡ ui − µi (1 + r) ≥ 0 so that both types of consumers are willing to borrower at interest rate r. Also seerelated discussions toward the end of Section 2.3.

15Usury laws prohibit lenders from charging borrowers excessively high interest rates on loans. In the U.S., manystates have established caps on the interest rate that lenders can charge for small dollar loans, such as payday andauto-title products. See, for instance, https://bit.ly/3mhJn2b for details.

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2.2 Lenders and Screening Ability

There are two risk-neutral competing lenders in the market. When a borrower applies for a loan,each lender conducts an independent creditworthiness test before deciding whether to make anoffer. We are interested in the case when one lender has a better screening ability than the other.As we emphasized in the introduction, screening ability includes both data availability and thedata processing technique/algorithm. We call one of the lenders a strong lender (denoted by s)and the other a weak lender (denoted by w). When it comes to the open banking applications innext sections, the two lenders will be a traditional bank and a fintech lender, which differ in theirscreening ability.

Following Broecker (1990), we assume that each lender receives an independent and privatesignal of a borrower’s type via a credit screening. Let Sj ∈ {H,L} denote lender j’s signal, wherej ∈ {s, w}. Lender j’s screening features a signal structure

P(Sj = H|h) = xhj , P(Sj = L|l) = xlj ,

with xis > xiw, i ∈ {h, l}. That is, when a borrower is of high type, lender j will receive a high signalH with probability xhj and a low signal L with probability 1−xhj ; when the borrower is of low type,the bank will receive a low signal L with probability xlj and a high signal H with probability 1−xlj .Hence

(xhj , x

lj

)represents lender j’s screening ability, which is publicly known and measures the

informativeness of lender j’s signal.16 In the following, we use high (low) signals and good (bad)signals exchangeably.

After receiving their private signals, the lenders update their beliefs about the borrower’s typeand make their loan offers rj ∈ [0, r] (if any) simultaneously. The borrower will choose the offerwith the lower interest rate. (When the two lenders offer the same deal, suppose the borrower willrandomly pick one offer, though the details of the tie-breaking rule do not affect our analysis.) Forsimplicity, we assume that the two lenders have the same funding costs which we normalize to 1.17

2.3 Simplifications

To reduce the number of parameters without losing our main insights, we simplify the model byassuming: (i) µl = 0 and µh = 1. That is, a low-type borrower never repays a loan while a high-typeborrower always repays;18 (ii) xhs = xhw = 1. That is, to either lender, a high-type borrower alwaysyields a good signal, so we have a “bad-news” signal structure. In other words, in our model a badsignal perfectly reveals that a borrower is of low type (and so no lender will lend to a borrower

16Broecker (1990) considers a general number of lenders with symmetric screening abilities, while we consider aduopoly case with asymmetric screening abilities.

17When it comes to open banking applications, we could alternatively assume that the fintech lender has a higherfinancing cost than the traditional bank. The fintech’s disadvantageous position in financing cost is a well-knownempirical regularity, because of their lack of cheap and stable funding sources like deposits. However, consideringasymmetric funding costs only complicates the analysis without adding significant economic insight given our focus.

18Although low-type borrowers always default, we assume that they prefer a cheaper loan, which can be justifiedif µl is slightly above zero.

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with a bad signal), while a good signal is inconclusive. From now on, we let

xs = xls > xw = xlw,

which summarizes the two lenders’ screening abilities.We assume that each lender is willing to lend to a borrower with a high signal H at the highest

possible interest rate r. The details of this assumption are as follows. For lender j, the chance thatit will see a high signal from a borrower is θ + (1 − θ)(1 − xj). Given µl = 1 and µh = 0, uponseeing a high signal it expects a repayment rate

θ

θ + (1− θ) (1− xj)= τ

τ + 1− xj,

where recall τ = θ1−θ . The lender is willing to lend at r = r if this expected repayment rate times

1 + r is greater than the cost 1. This requires

τr > 1− xj . (1)

This is easier to hold when there are more high-type borrowers in the population, i.e., a higher τ , orwhen the screening ability is better. This assumption, together with the bad-news signal structure,implies that in our model the only mistake lenders may make is lending to a low-type borrower.

Finally, we assume that any borrower of type i obtains a non-monetary benefit δi just fromgetting a loan. For high-type borrowers, they are left with some endogenous rent thanks to lendercompetition. We hence normalize δh to 0 for convenience as δh plays no role in our subsequentanalyses. We, however, set δl = δ > 0. In the context of small business loans, δ can be interpretedas the control rent of entrepreneurs from non-pledgeable income (see, for instance, Tirole, 2010),so that low-type borrowers who never succeed still care about the likelihood of getting a loan. Thismakes low-type borrowers’ welfare meaningful, and for our applications we think about the controlrent δ as relatively small.19

2.4 Equilibrium Characterization

We solve the model in closed-form in this section by fully characterizing the unique (mixed-strategy)equilibrium for credit market competition.

2.4.1 Preliminary analysis

LetpHH ≡ P (Ss = H,Sw = H) = θ + (1− θ) (1− xs) (1− xw)

19For the interpretation of consumption loans, δ then represents the low-type borrowers’ utility from consumingthe product given µl = 0. Following the discussion in footnote 14, we only need δ ≥ 0 so δ can be arbitrarily small.

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be the probability that both lenders observe a good signal from a borrower, and let

µHH ≡θ

pHH

be the probability of repayment of a borrower conditional on that. Similarly, denote by

pHL ≡ P (Ss = H,Sw = L) = (1− θ) (1− xs)xw

the probability that the strong lender observes a good signal but the weak one observes a badsignal, and by

pLH ≡ P (Ss = L, Sw = H) = (1− θ)xs (1− xw)

the probability that the stronger observes a bad signal but the weak one observes a good signal. Ineither case, the expected repayment probability is zero. Note that pLH > pHL given that xs > xw.

The credit market competition in our model has a flavor of common-value auctions. A lenderwins a borrower if it offers a better interest rate than its rival, or if the rival does not make anoffer at all, which happens when it sees a bad signal. Hence, winning the borrower brings somebad news—a winner’s curse. More precisely, suppose that the two lenders offer the same interestrate r ≤ r. Then the strong lender’s profit, for instance, is

pHH ×12 [µHH (1 + r)− 1]− pHL︸︷︷︸

winner’s curse

. (2)

When both observe a good signal from a borrower (which occurs with probability pHH), the stronglender wins the borrower with probability 1/2; when the strong lender observes a good signal butthe weak one observes a bad signal (which occurs with probability pHL), the former wins for surebut in that case the borrower must be of low type and so will never repay the loan.

Due to this winner’s curse, it is easy to see that in our model there is no pure-strategy equi-librium.20 As shown in the Online Appendix B.1, any mixed-strategy equilibrium in our model iswell behaved. Let mj , j ∈ {s, w}, be lender j’s probability that it makes an offer to a borrowerupon seeing a good signal. (As we will see, in the mixed-strategy equilibrium, the strong lenderwill always make an offer after seeing a good signal, while the weak lender will sometimes not makean offer.) Let Fj (r) ≡ Pr (rj ≤ r) be lender j’s interest rate distribution conditional on making anoffer, and as shown in Appendix B.1, the two lenders’ distributions must share the same supportwith common lower bound r (which will be specified below) and upper bound r. For our subsequentanalysis, it is more convenient to use the survival function F j(r) ≡ 1−Fj(r). Let πj be the lenderj’s equilibrium (expected) profit.

20It is impossible that the two lenders offer different interest rates; otherwise the lender offering a lower interestrate could always raise its interest rate slightly without losing any demand. If they charge the same interest rate andmake a nonnegative profit, then the first portion in (2) must be strictly positive, in which case each lender will havea unilateral incentive to undercut its opponent.

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In a mixed-strategy equilibrium, the strong lender’s indifference condition, when r ∈ [r, r], is

pHH[1−mw +mwFw (r)

][µHH (1 + r)− 1]− pHL = πs. (3)

When the strong lender offers interest rate r upon seeing a good signal, there are two possibilities: ifthe weak lender also observes a good signal (which occurs with probability pHH), the strong lenderwins if the weak one does not make an offer (which occurs with probability 1−mw) or if the weakone makes an offer but its interest rate is above r (which occurs with probability mwFw(r)); if theweak lender observes a bad signal instead (which occurs with probability pHL) and hence makesno offer, the borrower must be of low type and so the strong lender make a loss of 1. Similarly, theweak lender’s indifference condition is

pHH[1−ms +msF s (r)

][µHH(1 + r)− 1]− pLH = πw. (4)

Lemma 1. In any mixed-strategy equilibrium, the strong lender makes a strictly positive profitπs > 0 while the weak lender makes a zero profit πw = 0.

This is because the weak lender faces a higher lending cost due to its more severe winner’scurse (i.e., pLH > pHL). Given that there is no product differentiation, only the lender with thelower cost makes a positive profit. As we will see below, the strong lender’s profit actually equalspLH − pHL = (1− θ) (xs − xw).

2.4.2 Mixed-strategy competition equilibrium

Now we fully characterize the mixed-strategy equilibrium with πs > πw = 0. The strong lendermust always make an offer upon seeing a good signal (i.e., ms = 1) because of its strictly positiveprofit. Equation (4) then simplifies to

pHHF s (r) [µHH (1 + r)− 1]− pLH = 0. (5)

To make this equation hold for r close to r, we need Fs to have a mass point at the top. Letλs ≡ lim

r↑rF j (r) ∈ [0, 1) be the size of the mass point. (This also implies that the support of Fw

must be open at the top.) From (3) and (5), we can uniquely solve for all four endogenous variables(r, πs,mw, λs) and the two distributions. For notational convenience, we define

φ (r) ≡ pLHpHH [µHH (1 + r)− 1] = xs

τ1−xw r − 1 + xs

, (6)

which is F s(r) solving (5). Note that φ (r) depends on primitive parameters xw, xs, and τ .Denote by ∆ the gap in screening ability between the two lenders:

∆ ≡ xs − xw.

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One can also check that 1− φ (r) > 0 from assumption (1), which implies that φ (r) ∈ (0, 1). Thenthe mixed-strategy equilibrium is characterized as follows:

Proposition 1. The competition between the two lenders has a unique equilibrium in which:

1. the strong lender makes a profit πs = ∆1+τ and the weak lender makes a zero profit πw = 0;

2. the strong lender always makes an offer upon seeing a high signal (ms = 1), and its interestrate is randomly drawn from the distribution F s(r) = φ(r), which has support [r, r] withr = 1−xw

τ and has a mass point of size λs = φ (r) at r; and

3. the weak lender makes an offer with probability mw = 1−φ (r) upon seeing a high signal, andwhen it makes an offer the interest rate is randomly drawn from the distribution

Fw (r) = φ (r)− φ (r)1− φ (r) ,

which has support [r, r).

When τ goes to ∞ (i.e., when there is no default risk in the market), as expected the equilib-rium smoothly converges to the Bertrand equilibrium where both lenders offer r = 0. One usefulobservation is that for r ∈ [r, r), the two distributions satisfy

Fs (r) = mwFw (r) . (7)

Since mw = 1 − φ (r) < 1, this means the strong lender charges an interest rate higher than theweak lender in the sense of first-order stochastic dominance (FOSD). Intuitively, the weak lenderknows that its screening ability is relatively low and a good signal is not convincing enough todetermine that the borrower is of high type, and so it chooses not to lend sometimes. As a result,the strong lender sometimes acts as a monopoly credit supplier and charges a higher interest rate.

The following result reports how each lender’s screening ability and average credit quality affectthe competition.

Corollary 1. In the competition equilibrium,

1. when the screening ability gap ∆ increases or the average credit quality τ decreases, the stronglender’s profit (which is also the industry profit) increases; and

2. when the strong lender’s screening ability xs improves, or the weaker lender’s screening abilityxw deteriorates, or the average credit quality τ decreases, both lenders charge a higher interestrate in the sense of FOSD, and the weak lender makes an offer less frequently conditional onseeing a high signal.

This result suggests that the winner’s curse is the key driver of the degree of competition in ourmodel. When the screening ability gap ∆ is larger, the winner’s curse becomes more asymmetric

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between the two lenders, which softens competition. When the average credit quality τ is smaller,the winner’s curse for each lender becomes more severe, which also weakens competition.

Before leaving this section, we point out that Proposition 1 applies to the (generic) case ofxs > xw. However, the edge case xs = xw is slightly trickier. There are two asymmetric equilibria(which are the continuous limits of the equilibrium in Proposition 1), depending on which lenderalways makes an offer upon seeing a good signal. There is also a symmetric equilibrium whereneither lender always makes an offer upon seeing a good signal (i.e., ms = mw < 1). Lenders makea zero profit in any of these equilibria, but borrowers prefer the two asymmetric equilibria becausethere they are more likely to get a loan. For this reason, whenever this edge case matters, we focuson the asymmetric equilibria.

2.5 Borrower Surplus

The surplus of each type of borrowers is important for our subsequent analysis. Let Vi (xw, xs, τ)denote the expected surplus of an i-type borrower, i ∈ {h, l}, as a function of the two lenders’screening abilities and the average credit quality in the market.

A high-type borrower receives at least one offer (from the strong lender) and so always get aloan. The expected interest rate she pays is given by

(1−mw)E [rs] +mwE [min (rw, rs)] = r + (r − r)φ (r) , (8)

where φ (·) is defined as in (6). Here, when the weak lender does not make an offer, the borroweraccepts the strong lender’s offer; when both make offers, the borrower chooses the cheaper one. Theequality comes from using E [rs] = r +

∫ rr F s (r) dr and E [min (rw, rs)] = r +

∫ rr F s (r)Fw (r) dr.

Then a high-type borrower’s expected surplus is

Vh (xw, xs, τ) = (r − r) (1− φ (r)) . (9)

It is the high-type’s pecuniary payoff from the project and equals r net of the expected interestrate in (8).

Since a low-type borrower never pays back her loan, she cares only about the chance of gettinga loan. A low-type borrower will not receive any offer if the strong lender observes a bad signaland at the same time the weak lender either observes a bad signal or observes a good signal butdoes not make an offer. This occurs with probability xs [xw + (1− xw) (1−mw)]. Therefore, givenmw = 1− φ (r), a low-type borrower’s expected surplus is

Vl (xw, xs, τ) = δ [1− xs (xw + (1− xw)φ (r))] , (10)

where δ is the low-type’s non-monetary benefit from getting a loan as we have introduced beforefor the low-type borrower.

For our open banking applications, it is important to understand how each lender’s screening

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ability affects borrower surplus.

Proposition 2. Both types of borrower benefit from a higher average credit quality τ in the market.Regarding screening ability, both types of borrower suffer due to a higher screening ability of thestrong lender (i.e., a higher xs); high-type borrowers also benefit from a higher screening ability ofthe weaker lender (i.e., a higher xw), but low-type borrowers benefit from a higher screening abilityof the weaker lender (i.e., a higher xw) if and only if r

r < 1 +√xs.

The first result is straightforward from Corollary 1: A higher average credit quality lessens thewinner’s curse and so intensifies competition, and it also increases the chance that the weak lendermakes an offer upon seeing a good signal. The high-type benefit from both effects and the low-typebenefits from the second.

The intuition for the second result is as follows: When xs is improved, the screening abilitygap ∆ widens and this softens competition, and at the same time, the weak lender makes an offerless likely as it faces a more severe winner’s curse. The high type suffers due to both effects andthe low-type suffers due to the second. On the other hand, when xw is improved, the ability gap∆ shrinks and this intensifies competition, and at the same time the weak lender is more likely tomake an offer upon seeing a good signal (but for a low type borrower, the chance of generating agood signal declines). The high type benefits from both effects and the low type can be ambiguouslyaffected by the second effect.

In general, a change in screening ability brings about an informational effect, which enhancesthe screening efficiency; and a strategic pricing effect that affects the equilibrium interest rate aswell as the likelihood of a loan offer from the weak lender upon a good signal.21 These two effectscan be more clearly seen if we rewrite the borrower surplus in the parameter space {xw,∆, τ}, inwhich case xw is regarded as some base screening ability for both lenders, as formally stated inthe next corollary. When xw increases, both lenders’ screening abilities improve, and intuitivelythis should benefit the high type and harm the low type. On the other hand, a widening of thescreening ability gap ∆ worsens the winner’s curse problem, and this has a strategic pricing effectwhich lessens competition and impairs the welfare of borrowers.

Corollary 2. Once expressed as functions of {xw,∆, τ}, Vh increases while Vl decreases in thebase screening ability xw, and both Vh and Vl decrease in the screening ability gap ∆.

3 Open Banking: Credit Information Sharing

From now on, we consider a competition between a traditional bank (denoted by b) and a f intechlender (denoted by f). We assume that before open banking regulation, the bank is better atscreening borrowers because of its rich data from existing bank-customer relationships. More

21In our setup with µh = 1 and µl = 0, the first informational effect vanishes for the high-type borrowers since theyalways generate a good signal, and the interest rate effect in strategic pricing vanishes for the low-type borrowerssince they never repay the loan.

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specifically, let xj , j ∈ {b, f}, be lender j’s screening ability. We assume xf < xb before openbanking. After open banking, if the fintech has access to customer data from the bank, we assumethat its screening ability improves significantly to x′f so that it exceeds the traditional bank’s abilityxb. This is because, for example, the fintech is often equipped with more advanced technology tomake use of the data, or it has some additional customer information (e.g., from social media) thatcomplements the bank data. Therefore, in this section we assume

xf < xb < x′f . (11)

We aim to examine the welfare impacts of open banking and will mainly focus on the possibilitythat open banking has a perverse effect on borrowers. In the following, we first consider the casewhen the data sharing is mandatory (i.e., the data will be shared even without customers’ consent),and then consider the case of voluntary sign-up for data sharing as it works in practice.

3.1 Mandatory Sign-up

Suppose that all borrowers are required to sign up for open banking. This improves the fintech’sscreening ability, but it does not cause market segmentation since all borrowers have to share theirdata and so the lenders’ prior beliefs of the average credit quality remain unchanged. This is notthe practice of open banking regulation, but it is a useful benchmark.

Before open banking, the traditional bank is the strong lender and earns a positive profit∆

1+τ = xb−xf1+τ , and the fintech earns a zero profit; after open banking, the fintech becomes the

strong lender and earns a positive profit ∆′1+τ = x′f−xb

1+τ , and the bank earns a zero profit. Therefore,open banking increases industry profit if and only if it widens the screening ability gap betweenthe two lenders (i.e. if ∆′ > ∆).

Open banking increases the weak lender’s screening ability from xf to xb and may expandor shrink the screening ability gap between the two lenders. So its impact on borrowers is lessstraightforward. Open banking benefits borrowers of type i ∈ {h, l} if and only if Vi

(xb, x

′f , τ

)>

Vi (xf , xb, τ). (Recall that the first dependent variable in the borrower surplus function is the weaklender’s screening ability.) Proposition 2 implies that for a fixed xb, (i) Vh increases in xf < xb butdecreases in xf > xb, and (ii) Vl can vary with xf < xb non-monotonically but must decrease inxf > xb. Figure 1 depicts a numerical example of how Vh (Panel A) and Vl (Panel B) vary with xffor xb = 0.5.

Therefore, if xf is sufficiently close to xb before open banking and x′f is sufficiently above xb afteropen banking, both types of borrowers suffer from open banking. In other words, open banking isdetrimental to all borrowers if it causes a significantly larger new asymmetry between lenders. Itis also useful to think of the borrower surplus problem from the perspective of the base screeningability xw and the ability gap ∆ as in Corollary 2. Open banking improves the base screeningability, which benefits the high type but harms the low type. Hence, the high type will suffer fromopen banking only if it widens the gap (i.e. if ∆′ > ∆), in which case the low type must suffer from

15

Figure 1: Borrower Surpluses when Fintech Screening Ability Varies

Panel A plots the high-type borrower surplus Vh (xb, xf , τ) and Panel B plots the low-type borrowersurplus Vl (xb, xf , τ), both as functions of the fintech lender’s screening ability xf . The high-type borrowersurplus Vh (xb, xf , τ) is single-peaked at xf = xb (hence ∆ = 0) while Vl (xb, xf , τ) is hump-shaped in therange of xf < xb. Parameter values are r = 1, xb = 0.5, δ = 0.5, and τ = 1.1.

open banking while industry profit must be boosted.In our setup, high-type borrowers always get a loan in either regime, implying that open banking

is efficiency neutral to these borrowers. Low-type borrowers’ surplus is proportional to the chancethat they get a loan, and so whenever they suffer from open banking, it must be that these low-typeborrowers are less likely to get a loan, which improves the market efficiency if there is an efficiencyloss associated with them (which is the case as long as the low-type private benefit of receiving aloan δ < 1).

The above discussion is summarized in the following result:

Proposition 3. Compared to the regime before open banking,

1. for a fixed xb < 1, there exist xf < xb < x′f such that open banking with mandatory datasharing harms all borrowers if xf ∈ [xf , xb] and x′f ≥ x′f ; and

2. open banking with mandatory data sharing helps the fintech but harms the bank, and wheneverit harms all borrowers, it improves industry profit and market efficiency (if a low-type borrowergenerates an efficiency loss whenever she gets a loan).

Here we have focused on the potential perverse effect of open banking on borrowers. Of course,for other configurations of the parameters it is possible for open banking to benefit one or bothtypes of borrower.

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3.2 Voluntary Sign-up

We now turn to the more realistic and also more interesting case when signing up for open bankingis voluntary. Complying with the principles of GDPR in the EU (i.e., it is the customers, not thebank or the firm more generally, who own their personal data), the UK open banking regulationgives consumers the right to decide whether to allow fintech firms to access their personal bankingdata. But does this voluntary sign-up necessarily imply that consumers never get hurt? Consumers’sign-up decisions may reveal information on their credit quality, and this endogenous credit qualityinference will influence the lenders’ pricing strategies. As a result, it is ex ante unclear that openbanking with voluntary sign-up always helps consumers.

To facilitate our analysis where the equilibrium credit quality inference plays a key role, when-ever we study the voluntary sign-up equilibrium, we suppose that borrowers have heterogeneoussign-up costs for open banking. More specifically, a fraction ρ ∈ (0, 1) of borrowers, whom wecall “non--tech-savvy,” face an infinite sign-up cost and hence never sign up in equilibrium, whilethe remaining 1 − ρ of borrowers, whom we call “tech-savvy,” have a zero sign-up cost and theirsign-up decisions will be our focus. The sign-up cost is borrowers’ private information, and formodel parsimony, we assume it is independent of their credit quality type.

Although we label them based on “tech-savviness”, we emphasize that the distribution of sign-up costs captures a wide range of heterogeneity among open banking customers. For instance,some consumers are technology savvy, so that they not only “understand” the concepts of howtechnology works but also willing to “encompass” the utilization of such modern technology; someconsumers may deeply worry about the security of sharing their own data due to some unpleasantpersonal experience. Our analysis does not depend on the exact interpretation of the sign-up cost.

3.2.1 Sign-up decisions and equilibrium

Let σi ∈ [0, 1], for i ∈ {h, l}, be the fraction of i-type tech-savvy borrowers who choose to signup for open banking. Throughout, we use the two words “opt in” and “sign up” interchangeably(hence, “opt out” is equivalent to “not sign up”).

Consistent with open banking in practice, we assume that a borrower’s sign-up decision isobservable to both lenders.22 Then the two lenders compete in two separate market segments: onewhere borrowers sign up for open banking, and the other where borrowers do not. Let τ+ and τ− berespectively the lenders’ updated prior on the average credit quality in the two market segments.Specifically, τ+ ≡ Pr[h|sign up]

1−Pr[h|sign up] = τ · σhσl ,

τ− ≡ Pr[h|not sign up]1−Pr[h|not sign up] = τ · 1−(1−ρ)σh

1−(1−ρ)σl .(12)

Intuitively, when high-type tech-savvy borrowers are more likely to sign up for open banking, thelenders raise their estimate of the average credit quality in the opt-in segment but lower their

22The fintech of course observes the sign-up decision. It is also easy for the traditional bank to monitor borrowers’sign-up decisions since in practice the fintech needs to use the API provided by the bank to access the customer data.

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estimate in the other.Anticipating the equilibrium sign-up decisions in the population and the subsequent competition

outcome in each market segment, an i-type borrower’s sign-up decision is governed by:σi = 1, if Vi(xb, x′f , τ+) > Vi(xf , xb, τ−),

σi ∈ [0, 1] , if Vi(xb, x′f , τ+) = Vi(xf , xb, τ−),

σi = 0, if Vi(xb, x′f , τ+) < Vi(xf , xb, τ−).

(13)

If a borrower chooses to sign up, she will be classified in the market segment characterized by(xb, x′f , τ+) where the fintech becomes the strong lender; otherwise, she will be classified in themarket segment characterized by (xf , xb, τ−) where the fintech remains as the weak lender. Notealso that the surplus of an i-type non–tech-savvy borrower is Vi(xf , xb, τ−), since she never signsup for open banking.

An equilibrium with voluntary sign-up is a collection of{{σi} , {τ+, τ−} ,

{m+j , F

+j

},{m−j , F

−j

}},

together with some off-equilibrium beliefs whenever appropriate, so that (i) {σi} are the sign-updecisions of tech-savvy borrowers described in (13), (ii) {τ+, τ−} are the lenders’ updated prior ofthe average credit quality in each market segment as determined in (12), and (iii)

{m+j , F

+j

}and{

m−j , F−j

}are the lenders’ equilibrium pricing strategies described in Proposition 1 respectively

in the sign-up market segment characterized by (xb, x′f , τ+) and in the opt-out market segmentcharacterized by (xf , xb, τ−).

Although we do not intend to fully characterize all possible equilibria for the whole range ofparameters, we can rule out some types of equilibria immediately. First, it is impossible to havean equilibrium where signing up perfectly reveals the borrower type. This is the case when onlyhigh-type borrowers sign up (i.e., σl = 0 and σh > 0), or when only low-type borrowers sign up(i.e., σl > 0 and σh = 0). In the former case, a low-type borrower has an incentive to deviate andsign up so that she will be treated as a high-type borrower and so get a loan for sure; in the lattercase, a low-type borrower who signs up will never get a loan and so she has an incentive to deviate.Second, the following lemma rules out the equilibrium where both types of borrowers are indifferentabout whether to sign up or not. Intuitively, high-type borrowers are not afraid of a more precisescreening technology, and so they are more willing to sign up than low-type borrowers.

Lemma 2. If low-type tech-savvy borrowers are indifferent between signing up or not, then high-typetech-savvy borrowers must strictly prefer to sign up.

There is always an equilibrium in which nobody signs up for open banking, if we assign asufficiently unfavorable off-equilibrium belief to whoever signs up for open banking.23 But thisequilibrium is trivial in the sense that open banking has no impact at all on borrowers and lenders.In the following, we ignore this uninteresting equilibrium. Another possible simple equilibrium is

23If the condition in Proposition 3 holds (i.e. when mandatory sign-up makes all borrowers worse off), the off-equilibrium belief can be just the prior τ . This can be justified, for example, if there are some fintech lovers (withthe same credit quality distribution as the entire population) who always sign up for open banking.

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that all tech-savvy borrowers sign up for open banking (while lenders keep their prior τ in eachmarket segment), which requires Vi(xb, x′f , τ) ≥ Vi(xf , xb, τ) for both i = h, l. That is, this type ofequilibrium is sustained if both borrowers benefit from open banking with mandatory sign-up; inlater analysis, we will follow Proposition 3 below to rule out such equilibrium.

Only two possible types of equilibria remain: one with σh = 1 and σl ∈ (0, 1) in which thelenders’ updated prior in the sign-up market segment is τ+ > τ (i.e., signing-up is a signal favoringhigh type), and the other with σl = 1 and σh ∈ (0, 1) in which the lenders’ updated prior in thesign-up market segment is τ+ < τ (i.e., signing-up is a signal favoring low type). Intuitively, thefirst type of equilibrium is more sensible since high-type borrowers are more willing to face a fintechwith a higher screening ability. In the following, we explore the possibility that all borrowers sufferfrom open banking in this type of equilibrium.

3.2.2 Perverse impact of open banking

To sustain an equilibrium with σh = 1 and σl ∈ (0, 1), we need

Vh(xb, x′f , τ+) ≥ Vh(xf , xb, τ−) (14)

andVl(xb, x′f , τ+) = Vl(xf , xb, τ−), (15)

where τ+ > τ > τ−. The latter condition also implicitly requires that the opt-out market segmentbe active, i.e.,

τ−r > 1− xf . (16)

In this equilibrium, all low-type borrowers (regardless of their sign-up cost) must be harmed byopen banking since Vl(xf , xb, τ−) < Vl(xf , xb, τ) given τ− < τ . All high-type borrowers will sufferas well if

Vh(xb, x′f , τ+) < Vh(xf , xb, τ). (17)

The following result shows that such an equilibrium with all borrowers—regardless of their creditquality or tech-savviness—suffering from open banking exists for a range of primitive parameters.

Proposition 4. There exists a nonzero measure set of primitive parameters (as characterized inthe proof), so that

1. in the unique (nontrivial) equilibrium, (a) all tech-savvy high-type borrowers and a fractionof tech-savvy low-type borrowers sign up (i.e. σh = 1 and σl ∈ (0, 1)), and (b) all borrowersbecome worse off compared to the case before open banking; and

2. in the above equilibrium, compared to the case before open banking, the bank loses while thefintech gains, industry profit improves, and market efficiency improves as well (if a low-typeborrower generates an efficiency loss whenever she gets a loan).

19

For those borrowers who choose not to sign up, they are perceived to have a lower average creditquality than the whole population (τ− < τ) and so become worse off compared to the case beforeopen banking. For those who choose to sign up, they are viewed more favorably (τ+ > τ), but if thescreening ability gap becomes sufficiently large, they can still suffer from open banking. Of courseboth τ+ and τ− are endogenous in equilibrium, and Proposition 4 ensures that such parameterconfiguration exists so that open banking makes all borrowers worse off.

When both market segments are active, both lenders make a positive profit (the bank earnsfrom the opt-out market segment and the fintech earns from the sign-up market segment), but thebank earns less than before. When high-type borrowers also suffer from open banking, similarly asin the case of mandatory sign-up, the screening ability gap ∆′ must be sufficiently larger than ∆.This explains why the total industry profit must rise in this situation at the expense of borrowers,which is perhaps contrary to the original intention of open banking regulations.

4 Open Banking: Privacy and Targeted Loans

We so far have focused on sharing data on borrowers’ credit quality. However, the data that modernfinancial institutions process are multidimensional, and contain information on other aspects ofcustomer behavior as well. This section studies “privacy” concerns, i.e., some non-credit informationthat borrowers are reluctant to share with their lenders. Such extra information can be particularlyvaluable for fintech companies given their more advanced “big data” technology, but a certain typeof “precision marketing” based on such information could potentially hurt customers. Exactly outof this consideration, and guided by the open-data philosophy mentioned in the introduction, manyregulators around the world mandate consent from customers themselves when sharing their data.

We offer a tractable analysis of equilibrium open banking when non-credit privacy data sharingis intertwined with credit quality inference. We show that all borrowers could be worse off even ifthey control their own data (a similar result as we saw in the previous section); the endogenouscredit quality inference, as the backbone of the lending market competition, implies the sign-uppopulation is non-monotone in the degree of privacy concerns.

4.1 Consumer “Preferences” for Fintech Loans

Among many potential angles that one could take to explore “privacy,” we focus on the informationabout consumer/borrower preferences. This category of information complements well the infor-mation on credit quality studied in Section 3 which could be viewed as the information on loancosts.

Taking the baseline model in Section 2, suppose now that each borrower is subject to a privacyevent shock, so that with a probability ξ > 0 the borrower can take out loans only from the fintechlender. This privacy event, simply called the ξ-event, is independent of the borrower’s credit qualitytype. Before open banking, this privacy event is unobservable to both lenders. However, with openbanking, this event will be revealed to the fintech lender perfectly for borrowers who sign up for

20

open banking, so fintech lenders will be able to know exactly when borrowers are “locked in” tofintech loans. Albeit stark, our modeling of ξ-events is motivated by “event-based marketing,” andcaptures the idea that open banking enables fintech lenders to perform “precision marketing” bycombining the newly accessible borrower’s banking transaction records with some other existinginformation (e.g., the borrower’s social media data).24 Open banking hence helps fintech lendersinvade the borrower’s privacy.

In general, consumer privacy and precision marketing in our setting fall into two broad cat-egories. The first is when some borrowers strongly prefer fintech loans. For instance, when aconsumer shops on an e-commerce platform, she might have a strong preference for “immediacy”(i.e., buying a certain product immediately). If she needs to borrow but has no credit cards, inthis case fintech lenders often dominate traditional banks by processing loan applications muchfaster.25 With open banking, the transaction records from the borrower’s bank (which may containimportant information, say, on the borrower’s consumption habits), together with the borrower’sdigital footprint, often enable the fintech lender to better identify the event of demand immediacy.

The second category is when borrowers face a restricted set of available lenders in some circum-stances. For example, a borrower could be ineligible for bank loans sometime (e.g., because shehappens to be close to the bank’s borrowing limit), or she travels abroad and needs an emergencyloan in foreign currency (say for health insurances) unavailable from her bank. Open banking pro-vides such information to fintech lenders so that they can then target the borrower more precisely.

We assume that these ξ-events are realized “ex post”, after borrowers have made their sign-updecisions; this way, the belief updating with regard to the borrower’s opt-in/opt-out decision is onlyon credit quality, just like in Section 3. Our previous real-world examples are chosen to highlightthe idiosyncratic nature of these privacy events. In practice, customer borrowers often decide onceand for all whether to opt in or opt out of open banking when they start using the fintech services;a case-by-case decision likely involves a prohibitively high attention cost. Even if one can swiftlyopt out of open banking “without strings attached” as described in the Deloitte Insight survey inthe introduction, borrowers are unlikely to know exactly what data will be useful for the fintech,without mentioning that it might be too late to opt out as they have consented to sharing theirrecent banking history.

Given that the borrowers in the ξ-event can borrow from the fintech only, they are similar tothe “captured” consumers in Varian (1980). But our model offers some new economics due to thewinner’s curse embedded in the credit market competition. For instance, in contrast to the Varianmodel, we will show soon that the possibility of being captured does not always hurt a borrower,

24Precision marketing is a broader idea in retail business. Doug Shaddle, Director of Sales for UberMedia, once saidthat “the adoption of mobile technology is creating new data streams that can provide retailers with an unprecedentedamount of information about who their shoppers are and how to bring them further into the fold, ... to deliver theright offer, at the right time, to the right customer.” (See https://bwnews.pr/2FBXeA3.) Of course, broadly speaking,precision marketing could play a role for our study of credit information in Section 3 if the fintech, due to its superiortechnology, can classify borrowers into more categories after open banking and so tailor more personalized offers. Itis an interesting direction for future research.

25See, for instance, Fuster, Plosser, Schnabl, and Vickery (2019) for evidence that fintechs are faster at processingloans in the context of housing mortgages.

21

and also the fintech does not always earn a positive profit even with some captured borrowers.26

4.2 Competition Equilibrium before Open Banking

We first extend the baseline model in Section 2 to include the ξ-event and analyze the creditmarket competition before open banking. The resulting equilibrium serves as the benchmark forlater analysis.

4.2.1 The magnitude of ξ

The equilibrium structure crucially depends on the magnitude of ξ, i.e., the probability that aborrower can borrow from the fintech only. When

ξ ≤ φ (r) , xbτ

1−xf r − (1− xb)< 1, (18)

the fintech lender (who has a weaker screening ability) still makes a zero profit, and the equilibriumstructure is similar as in Proposition 1 (i.e., the baseline case of ξ = 0). When ξ > φ (r), however,both lenders earn a positive profit with a different equilibrium structure. In fact, by simply chargingthe maximum interest rate r (upon seeing a good signal), the fintech gains from borrowers whopass both screenings and are in their ξ-events, but loses from serving borrowers who are rejectedby the bank:27

ξ · pHH [µHH (r + 1)− 1]︸ ︷︷ ︸profit from ξ-events with two good signals

− pLH︸︷︷︸winner’s curse

= pLH

(ξ

φ (r) − 1). (19)

When ξ exceeds the critical value in (18), the gain dominates and hence the fintech with a sufficientmeasure of captured borrowers earns a positive profit.

4.2.2 Equilibrium characterization before open banking

When (18) holds, the bank always makes an offer upon seeing a good signal, while the zero-profitfintech makes an offer with probability mξ

f < 1, where we use the superscript “ξ” to indicate themodel with a possible ξ-event. The two lenders’ indifference conditions are similar to those in thebaseline:

pHH

(1− ξ)F ξb(r)︸ ︷︷ ︸win if beats bank

+ ξ︸︷︷︸win for sure

[µHH (r + 1)− 1]− pLH = πξf = 0︸ ︷︷ ︸fintech: zero profit

, (20)

26Also, with open banking our model is a variant of Varian (1980) where only one firm can identify its capturedconsumers and hence price discriminate accordingly. This scenario, which is quite natural in our context of creditmarket competition, is rarely considered in the literature on industrial organization.

27For the purpose of illustration, the borrower in the ξ-event is assumed to be screened by the bank (though nevertake a loan from it); this is without loss of generality because the fintech does not learn from the bank’s screeningresult.

22

and

(1− ξ)︸ ︷︷ ︸shrunk market size

{pHH

(1−mξ

f +mξfF

ξf (r)

)[µHH (r + 1)− 1]− pHL

}= πξb > 0︸ ︷︷ ︸

bank: positive profit

. (21)

When (18) does not hold, both lenders will make an offer for sure upon seeing a good signal, andso we will have mξ

f = 1 and πξf > 0 in the above two indifference conditions.Recall Vi (τ) in (9) and (10); we omit the screening ability variables since they are kept constant

in this section. The following proposition reports the details of the equilibrium.

Proposition 5. Before open banking, the equilibrium with possible privacy event can be character-ized as follows;

1. When ξ < φ (r), the fintech makes a zero profit πξf = 0 while the bank makes a profit πξb =(1− ξ) xb−xf1+τ > 0. The fintech adopts the same pricing strategy as in Proposition 1 withmξf = 1 − φ (r), and the bank’s pricing strategy is characterized by F

ξb (r) = φ(r)−ξ

1−ξ withmξb = 1. The borrower surpluses are:

V ξh (τ) = Vh (τ) , (22)

V ξl (τ) = Vl (τ)− ξδ (1− xb) (xf + (1− xf )φ (r)) . (23)

2. When ξ = φ (r), there exists a continuum of equilibria indexed by mξf ∈ [1− φ (r) , 1], which is

the fintech’s loan offer probability to a borrower with a good signal. Everything else is identicalto case (1), except for the low-type surplus which is given in Appendix A.9.

3. When ξ > φ (r), both lenders make positive profits. Upon seeing a good signal both lendersalways make an offer (i.e., mξ

f = mξb = 1), with interest rate distributions F ξb (r) = ξ

1−ξ ·φ(r)−φ(r)

φ(r) and F ξf (r) = ξφ(r) · φ (r). The borrower surpluses are

V ξh (τ) = (1− ξ)2

[r − (1− xb) (1− xf )

τ

]< Vh (τ) , (24)

V ξl = δ [(1− ξ) (1− xbxf ) + ξ (1− xf )] . (25)

The third case of a relatively large ξ > φ (r) is similar to the Varian-type model (with asymmetricsizes of captured consumers across firms). Thanks to its relatively large base of (potentially)captured borrowers, the fintech—despite its weaker credit screening ability—always extends loanoffers upon seeing a good signal and makes a positive profit. The larger the ξ, the more the capturedborrowers, and the higher interest rates from both lenders in the sense of FOSD.

The first case of a relatively small ξ < φ (r) is more “surprising.” We find that the fintech withrelatively few captured borrowers takes a pricing strategy that is independent of ξ—more precisely,it is the same as in the baseline ξ = 0. The bank, in contrast, prices more aggressively. Why? Astypical in a setting with a mixed-strategy equilibrium, the bank’s pricing strategy is determined by

23

the fintech’s zero-profit condition (20). But the zero-profit condition implies that the fintech mustlose from non-captured borrowers in equilibrium. As a result, the bank bids more aggressively (andearns less), so much so that high-type borrowers lose nothing from the presence of potential ξ-event;see Eq. (22). (This result differs from the case of large ξ just discussed above, or more generally,the Varian-type model.) On the other hand, because the potential ξ-event prevents borrowers fromtaking bank loans, this hurts low-type borrowers who only care about the chance of receiving aloan.28

Finally, in the knife-edge case (2), when ξ = φ (r) there exists a continuum of equilibria indexedby mξ

f ∈ [1− φ (r) , 1], the fintech’s probability of making an offer upon seeing a good signal. Thisexplains why low-type borrowers who care only about loan probabilities are affected by the fintech’spolicy. In this continuum of equilibria, mξ

f = 1−φ (r) corresponds to case (1) with ξ < φ (r), whilemξf =1 corresponds to case (3) with ξ > φ (r). This continuum of equilibria plays a role when we

analyze the model with voluntary sign-up.

4.3 Equilibrium Open Banking with Targeted Loans

We first solve the mandatory sign-up case to highlight the type-dependent incentives to opt in,and then characterize the equilibrium when sign-up is voluntary. To highlight the new role of openbanking in this section, we assume that the fintech lender’s screening ability on credit type remainsunchanged (i.e., x′f = xf ) after open banking.29 The fintech gains from open banking by takingadvantage of the borrowers’ data to extend targeted loans in the ξ-event.

4.3.1 Mandatory sign-up

Suppose that borrowers are mandated to sign up for open banking. To borrowers in their ξ-events,the fintech charges the monopolistic rate r whenever it sees a good signal. For borrowers in theirnon–ξ-events, lenders compete as in Proposition 1, leading a zero profit for the fintech. The fintech’sexpected profit hence is:

πξ,OBf = ξ ·

θr︸︷︷︸profit from high-type

− (1− θ) (1− xf )︸ ︷︷ ︸loss from low-type given H signal

= ξ · τr − (1− xf )1 + τ

> 0. (26)

Superscript “ξ,OB” indicates the ξ-event model under open banking.) For borrower surplus, in theξ-event, a high-type borrower is charged r (hence no rent left), while a low-type borrower receivesa loan given a good signal from the fintech (which occurs with probability 1− xf ). Therefore with

28The potential ξ-event hurts the low-type borrower, relative to the baseline model, only in the following scenario.The borrower receives a good signal from the bank (which occurs with prob. 1− xb) but the fintech does not makeany loan (which occurs with probability xf + (1− xf )φ (r), the fintech either receives a bad signal, or a good signalbut does not lend). This explains ξ (1− xb) (xf + (1− xf )φ (r)) in Eq. (23).

29For this reason, we have ignored the screening ability variables in the borrower surplus function in this section.

24

open banking and mandatory sign-up, the type-dependent borrower surpluses are:

V ξ,OBh (τ) = (1− ξ)Vh (τ) , (27)

V ξ,OBl (τ) = (1− ξ)Vl (τ) + δξ (1− xf ) , (28)

where Vi (τ) are in (9) and (10).By comparing them to Proposition 5, we have the next proposition on the impacts of open

banking (with mandatory sign-up):

Proposition 6. Compared to the regime before open banking,

1. there exists ξ ∈ (φ (r) , 1) such that high-type borrowers suffer from open banking with manda-tory sign-up if and only if ξ ≤ ξ, while low-type borrowers suffer if and only if ξ > φ (r).Therefore both types of borrower strictly suffer when ξ ∈

(φ (r) , ξ

); and

2. open banking with mandatory sign-up helps the fintech but (weakly) harms the bank.

The fintech benefits from open banking, as it now can price discriminate and offer targeted loansto exploit the borrowers in their ξ-events. The bank strictly suffers when ξ > φ (r), because inthat case after open banking, the fintech will compete more aggressively for non--ξ-event borrowers.(When ξ ≤ φ (r), the fintech adopts the same pricing strategy before and after open banking, andthat is why open banking has no impact on the bank.)

Open banking has an intriguing type-dependent impact on borrower surplus, which drives ouranalysis of the voluntary sign-up equilibrium in the next section. When ξ < φ (r) so that the fintechlender still earns zero profit before open banking, the high type suffer from open banking whichfacilitates the fintech to target their ξ-events. In comparison, the low type benefit since they nowreceive an offer for sure in the ξ-event if the signal is good (but before open banking in the sameevent, the fintech might not make offers as mξ

f < 1).When ξ > φ (r) so that the privacy concern is relatively large, the result concerning the low-

type surplus is reversed. Thanks to a sufficiently large number of captured borrowers, before openbanking the fintech lender makes a strictly positive profit and always offers a loan upon seeing agood signal. However, after open banking, the fintech can identify captured borrowers perfectly,and as a result it scales back in non--ξ-events (mξ,OB

f < 1 so it randomly drops out without makingoffers). Low-type borrowers thus prefer opting out of open banking.

For high-type borrowers, they could gain strictly from open banking when ξ > ξ; this is again incontrast to being harmed by open banking when ξ is small. To see this result, consider the limitingcase of ξ → 1. Before open banking, knowing that the fintech will be the de facto monopolist, bothlenders charge interest rates that converge to r. After open banking, the bank—knowing that thefintech can identify captured borrowers perfectly—offers an interest rate independent of ξ. Thisimplies that the high-type benefit from open banking.

Proposition 6 delivers a result that is parallel to Proposition 3 in Section 3 on credit quality datasharing: It is possible that both types of borrower strictly suffer from open banking with mandatory

25

sign-up. Just like in Section 3, this perverse welfare effect can hold even when borrowers voluntarilychoose to share their “privacy” data, as we show now.

4.3.2 Voluntary sign-up

Now we study the case with voluntary sign-up. As in Section 3, let ρ be the measure of non–tech-savvy borrowers with an infinite sign-up cost, which is independent of both credit quality type aswell as of the privacy event.

Recall that the fraction of the tech-savvy type-i borrowers who sign up for open banking isdenoted by σi ∈ [0, 1], and the updated priors of credit quality in the two market segments arerespectively τ+ for the opt-in market and τ− for the opt-out market as defined in (12). In thissection we further assume that ρ is sufficiently large:

ρτr > 1− xf . (29)

The condition says even if σh = 1 (all high-type tech-savvy borrowers opt in) while σl = 0 (alllow-type tech-savvy borrowers opt out), lenders still serve both segments thanks to a sufficientlyfavorable updated opt-out prior τ−, in light of condition (1).

Crucially, lender competition in the opt-out segment resembles that in Proposition 5 beforeopen banking, but the threshold value for ξ—which is φ (r; τ−)—is endogenous and depends on theupdated opt-out prior τ−. For this reason, we write the dependence of τ− of φ (r; τ−) explicitly;φ (r; τ−) is decreasing in τ−. The following proposition fully characterizes the unique equilibriumthat arises, when we vary ξ.

Proposition 7. With the potential privacy event and voluntary sign-up for open banking, we have

1. when ξ < φ (r; τ), there exists a unique equilibrium where no borrowers sign up, i.e., σh =σl = 0;

2. when φ (r; τ) < ξ < φ (r; ρτ), there exists a unique equilibrium where σh > σl > 0, so thatτ− = τ 1−(1−ρ)σh

1−(1−ρ)σl satisfies ξ = φ (r; τ−); and

3. when ξ > φ (r; ρτ), there exists a unique equilibrium where only high-type borrowers sign up,i.e., σh = 1 while σl = 0.

The case of small ξ < φ (r; τ). When ξ is sufficiently small, the unique equilibrium is thatnobody signs up for open banking. This explains why the average credit quality in the opt-outsegment is τ− = τ (i.e., the prior), and the lender competition in the opt-out segment falls intocase (1) of Proposition 5.

The intuition is as follows. As we have pointed out in Proposition 6, fixing the average creditquality, the low-type is more willing to opt in than the high-type. The high-type suffers from openbanking due to fintech exploitation of the privacy event, while the interest rate–insensitive low-typeon the contrary benefits from a greater chance of receiving a loan. It is in sharp contrast to Lemma

26

2 which concerns sharing credit quality data in Section 3. There, the high type naturally prefersa more precise screening technology (relative to the low type); while here, signing up for openbanking means exposing the high type to exploitation by the fintech charging a monopolistic rate(something that the low type do not care).

This gives rise to a “stigma” effect—akin to the one in the context of Fed’s discount window(e.g., Armantier, Ghysels, Sarkar, and Shrader, 2015)—of associating signing up with low creditquality. Then the low type would not sign up either because doing so would reveal their creditquality type. Consequently, the only equilibrium is nobody signing up.

The case of large ξ > φ (r; ρτ). When ξ is sufficiently large, the unique equilibrium is that onlyhigh-type (tech-savvy) borrowers sign up for open banking. This explains the equilibrium updatedopt-out prior τ− = ρτ , and the lender competition in the opt-out segment falls into case (3) ofProposition 5.

Again the endogenous credit quality inference is crucial, because the equilibrium is driven bylow-type borrowers always preferring to opt out. Eq. (25) in Proposition 5 shows that their opt-outsurplus V ξ

l is independent of τ−; in fact, V ξl achieves its upper bound because both the fintech (with

a sufficiently large measure of captured borrowers) and the bank always make an offer upon goodsignals. Opting in open banking exposes the low type to the risk of the fintech (as the weakerlender) not to make loans in their non–ξ-events. No low-type tech-savvy borrower will opt in, andthen the equilibrium inference is that whoever signs up must be a high-type borrower. We showthat this favorable credit quality inference is sufficient to convince the high type to always sign upfor open banking in equilibrium, despite the exposure of their privacy events.

The case of intermediate ξ ∈ (φ (r; τ) , φ (r; ρτ)). When ξ falls in the intermediate range, theunique equilibrium takes the form of the knife-edge case (2) in Proposition 5. There, the equilibriumsign-up populations of both (tech-savvy) types endogenously ensure that ξ = φ (r; τ−), and we pindown the fintech’s loan offering probability mξ

f from the two borrower’s indifference conditions.

4.4 Impact of Open Banking and Voluntary Sign-up

Our discussion is based on the following proposition. We first define the sign-up population to be

p (ξ) ≡ (1− ρ) [θσh (ξ) + (1− θ)σl (ξ)] . (30)

Proposition 8. The open banking sign-up population p (ξ) is single peaked at ξ ∈ (φ (r; τ) , φ (r; ρτ)).For all ξ ∈

[φ (r; τ) , ξ

], relative to the case before open banking, all borrowers are strictly worse

off; the fintech gains while the bank loses; and the financial industry gains under conditions givenin the proof.

27

Figure 2: Equilibrium Open Banking Sign-up Population and Borrower Surplus

Panel A plots the open banking sign-up population p (ξ) as a function of ξ (left scale); the updated opt-inand opt-out priors τ+ and τ− (right scale). The updated prior τ+ increases with ξ and diverges to ∞ asξ → φ (r; ρτ); therefore we cap it at 10. Panel B plots borrower surpluses as a function of ξ; the solid bluelines with crosses (dots) are surplus for the (non–) tech-savvy high-type borrowers, while solid red lines(squares) are surplus for the (non–) tech-savvy low-type borrowers. In the figure, φ (r; τ) = 0.18 andξ = 0.24. Parameter values are r = 1, xb = 0.8, xf = 0.5, δ = 0.5, ρ = 0.4, and θ = 0.7.

4.4.1 Sign-up population and privacy concerns

Panel A in Figure 2 illustrates the total sign-up population and the updated priors in equilibrium,as a function of ξ. When ξ increases, initially nobody signs up, p (ξ) = 0, with the updated opt-outprior τ− staying at the prior τ . Both types of tech-savvy borrowers start to sign up once ξ exceedsφ (r; τ), which takes a value of 0.18 in our numerical example. The updated opt-out prior τ− goesdown afterwards, while the updated opt-in prior τ+ in the opt-in segment always sits above theprior τ . The total sign-up population as shown peaks at ξ = 0.24 then goes down afterward; thisis because σh = 1 while σl decreases for ξ > ξ, explaining the pattern of updated priors in bothsegments. When ξ > φ (r; ρτ) = 0.48, as shown in Panel A the sign-up population p (ξ) remains at(1− ρ) θ > 0, which is the measure of tech-savvy high-type borrowers.

Our analysis hence generates a surprising comparative static result on the non-monotonic rela-tion between the equilibrium sign-up population p (ξ) and ξ. A casual thinking might suggest thatp (ξ) decreases with ξ, as ξ captures the borrowers’ privacy concern against data sharing. We showthat this casual thinking captures some economics, but only partially. In the scenario of small ξ,open banking allows fintech lenders to target high-type borrowers who are concerned about un-fair pricing, and in fact this stigma effect goes a long way to prevent everybody from signing upfor open banking, as discussed after Proposition 7. However, when the magnitude of the privacyconcern ξ is large, the opt-out incentive of low-type borrowers dominates the equilibrium creditquality inference, and eventually all tech-savvy high-type borrowers sign up for open banking inequilibrium.

Our paper hence sheds some light on the economics behind the slow adoption of the open

28

banking over time. Since the creation of the Open Banking Implementation Entity (OBIE) by theCompetition and Markets Authority (CMA) in the UK in 2016, the industry has witnessed littleenthusiasm from consumers.30 This lukewarm reception is often attributed to potential security-related privacy concerns caused by data sharing. Our analysis on the interaction between privacyand credit quality calls for a more careful examination of this conventional wisdom.

Taking a slightly different interpretation, our model also suggests that the adoption of openbanking might grow as the business models of the fintech lenders under consideration improves.Note, the parameter ξ measures the fraction of captured borrowers that can be potentially identifiedand targeted by fintech lenders. Therefore, another useful way to think about the magnitude ofξ as the proxy for the development of fintech businesses. The case of small ξ corresponds toan underdeveloped “challenger” fintech lender whose business model is in its relative early stage,lacking a clearly defined target market. In these small-ξ markets, nobody signs up for open bankingin equilibrium. Over time, our model predicts the sign-up population will grow, once fintech lendershave established their niche markets with more and more captured customers—e.g., young IvyLeague graduates who live in major metropolitan cities–and eventually become profitable afterlaunching certain successful differentiated products.

4.4.2 Welfare: the perverse effect of open banking

Proposition 8 shows that open banking could make all borrowers worse off even though they controltheir own data, and at the same time lead to a higher industry profit. This result follows irrespectiveof whether the data sharing concerns credit quality information as in Section 3, or some privacyinformation that facilitates making exploitative loans as studied here.

To understand the result regarding borrower surpluses,31 the perverse effect of open bankingemerges in the shaded green area as illustrated in Panel B in Figure 2. Consider ξ′ = φ (r; τ)+ε; wealready know from Proposition 6 that both types of borrowers suffer when sign-up is mandatory.The intuition for why the low type get worse off due to open banking is similar to Proposition 6 andquite straightforward. Before open banking, the fintech with a sufficiently large number of capturedborrowers lends aggressively (mξ

f = 1) upon seeing a good signal, benefiting the low type. Withopen banking and voluntary sign-ups, lender competition in the opt-out segment follows case (2)in Proposition 5; there, some endogenous mξ

f < 1 emerges to ensure the indifference condition ofborrowers regarding their sign-up decisions. This hurts the low-type borrowers, whose equilibriumvalue is the same regardless of whether they are tech savvy or not, opt-in or opt-out (Panel B inFigure 2).

Now we turn to high-type borrowers. For the tech-savvy high-type borrowers who choose to30For instance, see Warwick-Ching (2019), among others. The ongoing COVID-19 pandemic, which has forced

consumers and financial institutions alike to recognize the essential nature of digital interactions, offers a greatboost to the adoption of open banking. According to OBIE, over 2 million UK bank customers have connectedtheir accounts to trusted third parties by the end of September 2020, up from 1 million in January 2020. Seehttps://bit.ly/3kTvbvg.

31The intuition of the lender profit result is similar to the one given right after Proposition 6.

29

sign up, they suffer from the fintech’s exploitative targeted loans facilitated by open banking, asshown in Proposition 6 for ξ′ = φ (r; τ) + ε < ξ. Clearly, those who choose to opt out must beworse off as well thanks to the indifference equilibrium condition between opt-in and opt-out, andthe mechanism is the endogenous credit quality inference. To see this, the updated opt-out prior,i.e., τ− (ξ′) that solves ξ′ = φ (r; τ− (ξ′)), must be below the prior τ (as shown in Panel A in Figure8); this is because for the fintech to be break-even, we must have a lower updated prior for creditquality τ− to compensate for a larger measure of captured borrowers. But a lower τ− implies thathigh-type borrowers in the opt-out segment receive worse treatment from lenders, even if theychoose not to sign up for open banking.

5 Conclusion

As the volume of data created by the digital world continues to grow, customer data has evolvedinto a defining force in every business of banking including small business credit, consumptionloans, and retail services. Open banking regulation that requires banks to share their existingcustomers’ data with third parties—notably, fintech lenders—at customers’ requests can be viewedas an integral part of the broader “open economy” initiative, in which the data should be open tooutside third parties at the consent of customers who generate them.

We offer the first theoretical study on the consequence of letting borrowers control their owndata in an otherwise classic credit market competition between an incumbent traditional bank anda challenger fintech lender. Two kinds of data sharing by borrowers are explored: one concernstheir creditworthiness (i.e., information on lending cost), and the other their choice “privacy” (i.e.,information on customer preferences).

Although our results generally support the premise that open banking favors challenger fintechs,we highlight that the voluntary nature of data sharing is not sufficient to protect borrowers’ welfare.In both scenarios, we show the general existence of scenarios in which all borrowers are strictly worseoff (while the whole financial industry becomes more profitable), even for those who opt out of openbanking. This perverse effect is largely driven by the credit quality inference from borrower’s “sign-up” decisions, which is rooted in adverse selection as the backbone of credit market competition.Broadly, this effect is consistent with the information externality caused by consumer decisions,which poses a long-standing challenge to regulations on consumer protection in modern financialindustry.

There are a few other important issues on open banking that we leave for future research. First,traditional banks operate not only in the lending market but also in the deposit and payment servicemarket. Open banking affects their competition with fintech challengers in the latter market aswell, leading to another potential perverse effect on consumers. For instance, as the transactionaccount service provides the most valuable data for traditional banks, data sharing required byopen banking may dampen their incentives to compete in that market. Second, from a long-termperspective, should successful fintech giants also be required to share data back with traditional

30

banks? Third, we take open banking regulation as given; but is it better than the market mechanismwhere traditional banks act as data brokers and sell their data (upon customer consent) to fintechs?

31

ReferencesAbadi, Joseph, and Markus Brunnermeier, 2019, Blockchain economics, Available at SSRN.

Acquisti, Alessandro, Curtis Taylor, and Liad Wagman, 2016, The economics of privacy, Journalof Economic Literature 54, 442–92.

Ali, S Nageeb, Greg Lewis, and Shoshana Vasserman, 2020, Voluntary disclosure and personalizedpricing, in Proceedings of the 21st ACM Conference on Economics and Computation pp. 537–538.

Aridor, Guy, Yeon-Koo Che, and Tobias Salz, 2020, The economic consequences of data privacyregulation: Empirical evidence from gdpr, Discussion paper National Bureau of Economic Re-search.

Armantier, Olivier, Eric Ghysels, Asani Sarkar, and Jeffrey Shrader, 2015, Discount window stigmaduring the 2007–2008 financial crisis, Journal of Financial Economics 118, 317–335.

Banerjee, Priyodorshi, 2005, Common value auctions with asymmetric bidder information, Eco-nomics Letters 88, 47–53.

Berg, Tobias, Valentin Burg, Ana Gombović, and Manju Puri, 2020, On the rise of fintechs: Creditscoring using digital footprints, The Review of Financial Studies 33, 2845–2897.

Bergemann, Dirk, and Alessandro Bonatti, 2019, Markets for information: An introduction, AnnualReview of Economics 11, 85–107.

Biais, B., C. BisiÚre, M. Bouvard, and C. Casamatta, 2019, The blockchain folk theorem, TheReview of Financial Studies 32, 1662–1715.

Bouckaert, Jan, and Hans Degryse, 2006, Entry and strategic information display in credit markets,The Economic Journal 116, 702–720.

Broecker, Thorsten, 1990, Credit-worthiness tests and interbank competition, Econometrica: Jour-nal of the Econometric Society pp. 429–452.

Buchak, Greg, Gregor Matvos, Tomasz Piskorski, and Amit Seru, 2018, Fintech, regulatory arbi-trage, and the rise of shadow banks, Journal of Financial Economics 130, 453–483.

Dell’Ariccia, Giovanni, and Robert Marquez, 2004, Information and bank credit allocation, Journalof Financial Economics 72, 185–214.

, 2006, Lending booms and lending standards, The Journal of Finance 61, 2511–2546.

Di Maggio, Marco, and Vincent Yao, 2020, Fintech borrowers: Lax-screening or cream-skimming?,Available at SSRN 3224957.

Engelbrecht-Wiggans, Richard, Paul R Milgrom, and Robert J Weber, 1983, Competitive biddingand proprietary information, Journal of Mathematical Economics 11, 161–169.

Fuster, Andreas, Matthew Plosser, Philipp Schnabl, and James Vickery, 2019, The role of technol-ogy in mortgage lending, The Review of Financial Studies 32, 1854–1899.

Hausch, Donald B, 1987, An asymmetric common-value auction model, The RAND Journal ofEconomics pp. 611–621.

32

Hauswald, Robert, and Robert Marquez, 2003, Information technology and financial services com-petition, The Review of Financial Studies 16, 921–948.

He, Zhiguo, and Lin William Cong, 2019, Blockchain disruption and smart contracts, The Reviewof Financial Studies 32, 1754–1797.

Ichihashi, Shota, 2020, Online privacy and information disclosure by consumers, American Eco-nomic Review 110, 569–95.

Jones, Charles I., and Christopher Tonetti, 2020, Nonrivalry and the economics of data, AmericanEconomic Review 110, 2819–58.

Kagel, John H, and Dan Levin, 1999, Common value auctions with insider information, Economet-rica 67, 1219–1238.

Liu, Zhuang, Michael Sockin, and Wei Xiong, 2020, Data privacy and temptation, Discussion paperNational Bureau of Economic Research.

Marquez, Robert, 2002, Competition, adverse selection, and information dispersion in the bankingindustry, The Review of Financial Studies 15, 901–926.

Milgrom, Paul, and Robert J Weber, 1982, The value of information in a sealed-bid auction, Journalof Mathematical Economics 10, 105–114.

Pagano, Marco, and Tullio Jappelli, 1993, Information sharing in credit markets, The Journal ofFinance 48, 1693–1718.

Parlour, Christine A, Uday Rajan, and Haoxiang Zhu, 2020, When fintech competes for paymentflows, Available at SSRN.

Rajan, Raghuram G, 1992, Insiders and outsiders: The choice between informed and arm’s-lengthdebt, The Journal of finance 47, 1367–1400.

Sharpe, Steven A, 1990, Asymmetric information, bank lending, and implicit contracts: A stylizedmodel of customer relationships, The Journal of Finance 45, 1069–1087.

Srinivas, Val, Jan-Thomas Schoeps, and Aarushi Jain, 2019, Executing the open banking strategyin the united states, Deloitte Insights.

Stole, Lars A, 2007, Price discrimination and competition, Handbook of Industrial Organization 3,2221–2299.

Tang, Huan, 2019, Peer-to-peer lenders versus banks: substitutes or complements?, The Review ofFinancial Studies 32, 1900–1938.

Thakor, Anjan V, 1996, Capital requirements, monetary policy, and aggregate bank lending: theoryand empirical evidence, The Journal of Finance 51, 279–324.

Tirole, Jean, 2010, The theory of corporate finance (Princeton University Press).

Varian, Hal R, 1980, A model of sales, The American Economic Review 70, 651–659.

Vives, Xavier, 2019, Digital disruption in banking, Annual Review of Financial Economics 11,243–272.

33

Von Thadden, Ernst-Ludwig, 2004, Asymmetric information, bank lending and implicit contracts:the winner’s curse, Finance Research Letters 1, 11–23.

Warwick-Ching, Lucy, 2019, Open banking: the quiet digital revolution one year on, FinancialTimes.

A Appendix A

A.1 Notation Summary

Table 1: Notation Summary

Notation Definition and Meaning Characterization

θ Probability of high-typeτ Likelihood ratio of high-type τ = θ

1−θ

ρ Proportion of non–tech-savvy borrowersµi, i ∈ {h, l} Probability that a high/low-type repays µh = 1δi, i ∈ {h, l} Borrower’s private benefit of receiving a loan δh = 0, δl = δ > 0Vi(xw, xs, τ) Borrower i’s surplusj ∈ {b, f, s, w} Lender: traditional bank, or f intech; strong, or weakSj ∈ {H,L} Signal of lender j, is H or L

xj Screening ability of lender j in “bad news” structure P(Sj = L|l) = xj

pHH , pHL, pLH , pLL Probabilities of lender signalsµHH , µHL, µLH , µLL Probabilities of repayment for borrowers with given signals

r Upper bound of net interest rate (exogenous)r Lower bound of net interest ratemj Probability that lender j grants a loan given Sj = H

rj Net interest rate offered by lender jFj(r);F j(r) CDF of rj ; survival function of Fj(r) F j(r) = 1− Fj(r)

λj The mass point of Fj (r) at r λj = limr↑rF j (r)

πj Lender j’s profitφ(r) Eq (6)∆ Gap of screening ability ∆ = xs − xwx′f Screening ability of fintech after open banking in Section 3

σi, i ∈ {h, l} Proportion of type i tech-savvy borrowers who opt inτ+, τ− Updated prior of borrowers who opt in (+), and who opt out (-)ξ Probability of privacy eventp(ξ) Population of opt-in borrowers in Section 4

A.2 Proof of Lemma 1

Proof. Suppose first that πs, πw > 0 in equilibrium. Then both lenders make an offer for sure uponseeing a good signal (i.e. ms = mw = 1). From the two lenders’ indifference conditions, we can

34

see that as r ↑ r, at least one of F s(r) and Fw(r) will be zero since it is impossible that bothdistributions have a mass point at r = r. Thus, at least one of the lenders will make a negativeprofit, which is a contradiction.

Suppose then πw ≥ πs = 0. Then at r = r, we must have Fw(r) = F s(r) = 1, and so we needpLH ≤ pHL to make both indifference conditions hold. But as we pointed out before this cannotbe true given xs > xw. Therefore, the only remaining possibility is that πs > πw = 0.

A.3 Proof of Proposition 1

Proof. We have known the strong lender’s distribution is F s(r) = φ(r). From F s(r) = 1, we solver = (1− xw)/τ , which is less than r given condition (1). The size of Fs’s mass point is λs = φ(r),which is less than 1 given condition (1). Letting r = r in (3) yields πs = pLH−pHL = (1−θ)∆ = ∆

1+τ ,and letting r = r in (3) yields 1−mw = φ(r). Finally, Fw(r) is solved from (3).

A.4 Proof of Corollary 1

Proof. (i) Given πs = ∆1+τ ,the result concerning profit is obvious.

(ii) For any given r ∈ [r, r], it is easy to see that φ(r) defined in (6) increases in xs, decreasesin xw, and decreases in τ . So the claims follow immediately on the strong lender’s interest ratedistribution and the weak lender’s probability of making an offer upon seeing a good signal. To seethe result concerning the weak lender’s interest rate distribution, notice that the derivative of

Fw(r) = xs(1− xw)τr − (1− xw)

r − rr − (1−xs)(1−xw)

τ

with respect to xs is proportional to

τr − (1− xw) ≥ 0,

where the inequality is because r = (1− xw)/τ . It is easy to see that Fw(r) decreases in both xw(as the numerator decreases in xw and the denominator increases in xw) and τ (as the denominatorincreases in τ).

A.5 Proof of Proposition 2

Proof. Result (i) is immediate from Corollary 1. A higher τ induces both lenders to offer lowerinterest rates (in the sense of first-order stochastic dominance) and also induces the weak lender tomake offers more likely upon seeing a good signal. This benefits both types of borrowers.

35

The result concerning the impact of xs in (ii) is also immediate from Corollary 1. A higher xsinduces both lenders to charge higher interest rates and also induces the weak lender to make offersless likely upon seeing a good signal. This harms both types of borrowers.

When xw increases, we know from Corollary 1 that interest rates go up and the weak lenderoffers loans more likely upon seeing a good signal, and so the high-type must become better off. Butnow the weak lender receives a high signal less likely from a low-type borrower, and this negativelyimpacts the low-type borrowers. A straightforward calculation of the derivative of Vl with respectto xw yields the cut-off result.

A.6 Proof of Corollary 2

Proof. The result concerning the impact of ∆ is immediately from Proposition 2 since for a fixedxw increasing ∆ is the same as increasing xs.

The result concerning the impact of the base screening ability xw is less straightforward. Fornotational simplicity, in the proof let x = xw represent the base screening ability. Notice that

Vh (x,∆, τ)− δ = r

(1− 1− x

rτ

)[1− φ(r)] ,

whereφ(r) = x+ ∆

τ1−xr − 1 + x+ ∆ .

Its derivative with respect to x equals

[rτ − (1− x)] [∆ (1− x+ rτ) + 2rτx]

τ[∆ (1− x)− (1− x)2 + rτ

]2 > 0,

where the inequality is from 0 < x < 1 and Assumption 1 which implies rτ − (1− x) > 0.For the low-type borrowers,

1δVl(x,∆, τ) = 1− (x+ ∆) [x+ (1− x)φ(r)] ,

Its derivative with respect to x equals

− [rτ − (1− x)] [∆ (1− x+ rτ) + 2rτx][∆ (1− x)− (1− x)2 + rτ

]2 < 0.

36

A.7 Proof of Lemma 2

Proof. Let us define the φ function and the lower bound of the interest rate distribution in eachmarket segment as follows:

φ+ (r) = φ(r;xb, x′f , τ+), φ− (r) = φ(r;xf , xb, τ−),

and

r+ = 1− xbτ+

, r− = 1− xfτ−

.

When low-type borrowers are indifferent between signing up or not, from Vl defined in (10) weknow

x′f [xb + (1− xb)φ+ (r)] = xb [xf + (1− xf )φ− (r)] .

Given x′f > xb > xf and φ+ (r) , φ− (r) ≤ 1, we deduce that

xb + (1− xb)φ+ (r) < xf + (1− xf )φ− (r) ≤ xb + (1− xb)φ− (r) ,

and soφ− (r) > φ+ (r) . (31)

Using the expression for the φ function, we get

φ− (r) = xbrr−− (1− xb)

> φ+ (r) =x′f

rr+− (1− x′f )

>xb

rr+− (1− xb)

,

where the second inequality used x′f > xb and rr+

> 1. Hence,

r− > r+. (32)

Then from (31), (32) and Vh defined in (9), we derive

Vh(xb, x

′f , τ+

)− δ = (r − r+) (1− φ+ (r)) > Vh (xf , xb, τ−)− δ = (r − r−) (1− φ− (r)) ,

i.e. high-type borrowers with a zero sign-up cost must strictly prefer to sign up.

A.8 Proof of Proposition 4

Proof. (i) Here we prove that there is a non-empty set of primitive parameters such that (14)-(17) hold. (We relegate to the online appendix the detailed characterization of the range of theparameters and the condition for uniqueness.)

37

First, by continuity we can focus on the edge case with xb = xf . (Our argument below continuesto work when xb and xf are sufficiently close to each other.)

Second, given Vh decreases in the strong lender’s screening ability and x′f > xb, (17) must holdif τ+ is sufficiently close to τ . This can be achieved if we let σl be sufficiently close to 1.

Third, we choose τ− < τ so that the equality of (16) holds. (This is feasible given (1). Bycontinuity the argument also works for a slightly bigger τ−, in which case (16) holds.) The advantageof choosing τ− in this way is that we have Vh(xf , xb, τ−)− δ = 0 in the opt-out market segment sothat (14) must hold. When σh = 1, we have

τ− = τ · ρ

1− (1− ρ)σl.

Then for any τ− < τ and σl ∈ (0, 1), we must be able to find a ρ ∈ (0, 1) which solves the aboveequation.

Finally, we need to ensure that (15) holds for some parameters. The remaining parameter wecan choose is x′f . When the equality of (16) holds, one can check that Vl(xf , xb, τ−)/δ = 1 − xb.Then (15) requires

x′f

xb + (1− xb)x′f

rτ+1−xb − 1 + x′f

= xb.

Notice that when τ+ = τ , given (1), there exists ε > 0 such that the above equation has a solutionx′f ∈ (xb + ε, 1).32 The same argument works if τ+ is sufficiently close to τ . That is, for a τ+ ≈ τ

(or σl ≈ 1) chosen in the second step, the above equation has a solution x′f bounded away from xb

so that (15) holds.(ii) Before open banking, the bank earns π0

b = ∆1+τ and the fintech earns π0

f = 0. After openbanking, let n+ and n− be the measure of consumers who sign up and who do not, respectively.(They satisfy n+ + n− = 1.) Notice that we must have n+ (1− θ+) + n− (1− θ−) = 1 − θ, whereθ+ and θ− are respectively the fraction of high-type borrowers in each market segment. This isequivalent to

n+1 + τ+

+ n−1 + τ−

= 11 + τ

. (33)

In the sign-up market segment, the two lenders’ profits are respectively

π+b = 0, π+

f = n+∆′

1 + τ+.

32

Proof. This is because when x′f = 1, the left-hand side must be strictly greater than xb; when x′f = xb + ε, theleft-hand side must be strictly less than xb + ε for some ε > 0.

38

In the opt-out market segment, the two lenders’ profits are respectively

π−b = n−∆

1 + τ−, π−f = 0.

It is clear that the fintech earns a higher profit than before, while the bank’s profit drops as

π0b = ∆

1 + τ> π+

b + π−b = n−∆

1 + τ−,

where the inequality used (33).Industry profit goes up if and only if

π+f + π−b = n+

∆′

1 + τ++ n−

∆1 + τ−

> π0b = ∆

1 + τ.

Given (33), this is the case if ∆′ > ∆, which must be true in our equilibrium where the high-typeborrowers who sign up suffer from open banking. (This is because from Corollary 2, we know thatVh increases in the base screening ability and the average credit quality but decreases in the abilitygap. In the sign-up market segment, the base ability improves from xf to xb and the average creditquality improves from τ and τ+, and so the high-type borrowers become worse off only if ∆′ > ∆.)

The result concerning market efficiency follows from the same argument as in the case of com-pulsory sign-up.

A.9 Proof of Proposition 5

Before open banking, fintech cannot condition its strategy on the ξ-event. Similar results as Lemma3 apply here: when making an offer, lenders randomize over common support

[rξ, r

], and at most

one of them can have a mass point at the top r = r.

The case of ξ < φ (r):

Proof. From the discussion of ξ’s critical value (see Equation 19), if ξ < φ (r), fintech’s profit in theξ-event is dominated by the winner’s curse in the non--ξ-event when evaluated at r = r. Hence, bythe same argument in Proposition 1, one lender makes zero profit and randomly drops out uponseeing a good signal, and the other lender earns a positive profit, always makes an offer and has amass point at r = r, such that both lenders to be willing to offer at r = r. Similar to the argumentin Lemma 1, if πξf > πξb = 0, then (1− ξ)πξf > πξb , which further implies pLH < pHL, contradiction.Therefore, following Proposition 1, there exists a unique mixed strategy equilibrium with fintechrandomly dropping out mξ

f < 1 and traditional bank’s mass point λξb at the top.The equilibrium characterization largely follows from the baseline model. Recall φ(r) = xb

τ1−xf

r−(1−xb)

39

given in (6). Then (20) yields:

Fξb(r) = φ (r)− ξ

1− ξ = 11− ξ

(xb

τ1−xf r − (1− xb)

− ξ),

which is well defined when ξ ≤ φ (r) < 1; F ξb has a mass point at r = r with the size of

λξb = φ (r)− ξ1− ξ .

The traditional bank’s indifference condition (21) is the same as in the baseline, so fintech’s strategymust be the same as in the baseline: upon seeing a good signal, it makes an offer with probabilitymξf = 1− φ (r), and the offer randomizes over

[rξ, r

]according to

Fξf (r) = φ (r)− φ (r)

1− φ (r) .

We now derive the borrower surplus. As the mixed-strategy equilibrium here only differs fromthe baseline case in F ξb (r) = φ(r)−ξ

1−ξ (and λξb = φ(r)−ξ1−ξ ), it is convenient to illustrate the borrower

surplus as the benchmark surplus Vi plus a wedge due to the ξ-event. The high-type borrowerscare about the expected interest rate, so

V ξh (τ) =

(1− ξ + ξmξ

f

)r −

{(1− ξ) ·

[(1−mξ

f )E[rξb ] +mξfE[min{rξb , r

ξf}]]

+ ξ ·mξf · E[rξf ]

}︸ ︷︷ ︸

expected interest rate

where the second term in the curly bracket corresponds to the ξ-event in which there is only onelender. Plugging in F ξj (r) and mξ

f yields V ξh (τ) = Vh (τ) . Low-type borrowers only care about the

probability of receiving a loan, so

V ξl (τ) = (1− ξ)Vl (xf , xb, τ) + ξ (1− xf ) (1− φ (r)) δ.

In the 1− ξ event, the equilibrium differs from baseline equilibrium only in F ξb (r), and thus a lowtype who does not care about pricing has the same surplus Vl (xf , xb, τ) as in baseline; in the ξ-event, the fintech is the only lender, and the borrower only receives the loan when (wrongly) testedwith H (probability 1− xf ) and the fintech does make the offer (probability mξ

f = 1− φ (r)).

The case of ξ = φ (r):

Proof. When ξ = φ (r), from lenders’ indifference conditions (20) and (21), we know lender profitsπξb , π

ξf , lowest interest rate rξ, and bank pricing distribution F

ξb(r) are the same form as when

ξ < φ (r). At r = r, the size of bank’s mass point shrinks to λξb = 0 exactly, and it follows that thefintech may also yield borrowers to the traditional bank (to make the latter participate) by a masspoint at r, in addition to randomly dropping out upon H as in the case of ξ < φ (r). Hence, there

40

exists a continuum of equilibria indexed by mξf ∈ [1− φ (r) , 1] that satisfy 1−mξ

f +mξfλ

ξf = φ (r).

Accordingly λξf = 1− 1−φ(r)mξf

, and F ξf (r) = 1− 1−φ(r)mξf

from rescaling (still 1−mξf +mξ

fFξf (r) = φ (r)

as in ξ < φ (r)). This completes the characterization of the mixed strategy equilibrium.The choice of mξ

f affects the probability of receiving the loan and hence low-type’s surplus,while high type still earns V ξ

h (τ) = Vh (τ). Specifically,

V ξl (τ) = δ

{1− xb

[xf + (1− xf )

(1−mξ

f

)]}.

The case of ξ > φ (r):

Proof. Similar to Varian (1980), the unique equilibrium is a mixed-strategy one on common support[rξ, r

]; rξ will be shown to be different from other cases shortly. First, we argue that both lenders

have positive profits and always make an offer upon seeing a good signal, so mξj = 1 for j ∈ {b, f}.

To see this, one feasible strategy for the fintech is to always offer r = r upon seeing a good signal,and the associated profit is no less than

pHHξ [µHH (r + 1)− 1]− pLH = (1− θ)xb (1− xf )

ξ

φ (r)︸ ︷︷ ︸>1

−1

> 0.

To make the traditional bank willing to offer at r = r, fintech has a mass point λξf at r and thetraditional bank is open at r. The traditional bank must also make positive profit πξb > 0 due tobetter screening ability.33

The fintech’s indifference condition is

r ∈(rξ, r

): πξf = pHH

[ξ + (1− ξ)F ξb (r)

][µHH (r + 1)− 1]− pLH , (34)

Evaluating (34) at r = r yields the fintech’s profit

πξf = ξ · τr − (1− xf )1 + τ

− (1− ξ) · xb (1− xf )1 + τ

, (35)

which allows us to solve for rξ (as the fintech is earning πξf at rξ as well):

rξ = ξr + (1− ξ) (1− xb)(1− xf )τ

. (36)

Note that the lower bound here is higher than that in the baseline, rξ > r, because at lower boundinterest rate the fintech serves all borrowers tested with Sf = H in both cases but here πξf > 0.

33To see this, consider when a lender posts r = rξ and gets to serve all borrowers tested with HH. Then adjustingfor market size, the traditional bank suffers from less serious winner’s curse.

41

Lastly, the fintech is indifferent across r ∈[rξ, r

), implying

Fξb (r) = ξ

1− ξ ·φ (r)− φ (r)

φ (r) .

The bank’s indifference condition is

r ∈(rξ, r

): πξb = (1− ξ)

{pHHF

ξf (r) [µHH (r + 1)− 1]− pHL

}, (37)

Using this condition at r = rξ, we have bank profit

r = rξ : πξb = (1− ξ) pLH(

ξ

φ (r) −pHLpLH

);

The bank’s indifference condition across r ∈[rξ, r

)pins down the fintech’s strategy

Fξf (r) = ξφ (r)

φ (r) ,

with the mass point λf = ξ. Note that F ξf (r) strictly increases in ξ, so with a larger ξ the fintechoffers loans at higher interest rates in the sense of first order stochastic dominance.

As for borrower surplus, a high-type borrower always receives a loan and cares about theexpected interest rate,

V ξh (τ) = r −

[(1− ξ)E

[min

{rξb , r

ξf

}]+ ξE

[rξf

]]= (1− ξ)2

[r − (1− xb) (1− xf )

τ

];

a low-type borrower receives a loan when in the ξ-event she is tested H with the fintech, or whenotherwise she is tested H with at least one of the lenders,

V ξl (τ) = δ [ξ (1− xf ) + (1− ξ) (1− xbxf )] .

In addition, we show that high types are worse off due to the very likely privacy event,i.e. V ξ

h (τ) < Vh (τ) when ξ > φ (r). Recall that the expected interest rate in the baseline is

42

mfE [min {rb, rf}] + (1−mf )E [rb] = r +∫ rr φ

2 (r) dr, and the expected interest rate here is

(1− ξ)E[min

{rξb , r

ξf

}]+ ξE

[rξf

]= rξ + ξ2

φ2 (r)

∫ r

rξφ2 (r) dr

= r +∫ rξ

rdr︸ ︷︷ ︸

rξ>r

+ ξ2

φ2 (r)︸ ︷︷ ︸≥1

∫ r

rξφ2 (r) dr

> r +∫ rξ

rφ2 (r) dr +

∫ r

rξφ2 (r) dr

= r +∫ r

rφ2 (r) dr.

Note that there is a discontinuous downward jump in V ξh (τ) at the threshold ξ = φ (r): for a

smaller ξ the ξ-event does not affect borrower surplus but a larger ξ makes her worse off.

A.10 Proof of Proposition 6

Proof. 1. When ξ ≤ φ (r), the fintech earns a zero profit before open banking but a positive profitafter, and the bank makes the same profit in either case. When ξ > φ (r), from (26) and (35) it isimmediate to see that the fintech benefits from open banking; while the bank suffers as

πξb = (1− ξ){pHH

[µHH

(rξ + 1

)− 1

]− pHL

}> πξ,OBb = (1− ξ) {pHH [µHH (r + 1)− 1]− pHL} ,

where rξ > r as shown in Proposition 5.2. When ξ ≤ φ (r), it is straightforward to check that V ξ,OB

h (τ) < V ξh (τ) in (22) and

V ξ,OBl (τ) > V ξ

l (τ) from fintech’s offering probability in the ξ-event. When ξ > φ (r), the lowtype suffer from open banking because by comparing (25) and (28), we have

V ξl = δ (1− ξ) (1− xbxf ) + δξ (1− xf ) > (1− ξ)Vl (τ) + δξ (1− xf ) = V ξ,OB

l (τ) , (38)

where the inequality holds as δ (1− xbxf ) is greater thanVl (τ) in (10). (The equality will hold ifτ →∞.)34 The high type suffer from open banking if

V ξh (τ) = (1− ξ)2

[r − (1− xb) (1− xf )

τ

]> (1− ξ)Vh(τ) = V ξ,OB

l (τ) ,

which is equivalent to

(1− ξ) (r − r + xbr) > Vh(τ) = (r − r) (1− φ (r)) .34In the knife edge case open banking may hurt or benefit low-type depending on the equilibriummξ

f ∈ [1− φ (r) , 1] .

43

This holds if and only if ξ is below some threshold ξ ∈ (φ (r) , 1).3. This result directly follows from result 2).

A.11 Proof of Proposition 7

For notational convenience, we denote by ∆V ξ,OBi , V ξ,OB

i (τ+)−V ξi (τ−) the i-type’s incentive

to sign up. The sign-up equilibrium is a collection of tech-savvy borrowers’ sign-up decisions{σi}, and beliefs about the average credit quality in each market segment {τ−, τ+}, such that a){τ−, τ+}are determined by the Bayes’ rule and characterized in (12); b) {σi}satisfy borrowers’incentive compatibility conditions that are similar to (13) with surplus V ξ

i (τ−) for not signingup and V ξ,OB

i (τ+) for signing up, given lenders’ pricing strategies{mξ,OBj+ , λξ,OBj+ , F ξ,OBj+

}and{

mξ,OBj− , λξ,OBj− , F ξ,OBj−

}respectively for borrowers who opted in and who opted out.

Contrary to Subsection 4.3.1, now the threshold of ξ which decides the lender strategy in the opt-out segment is endogenous and depends on τ− (for borrowers who signed up, ξ does not affect thestructure of lender competition). We first characterize the case with ξ < φ (r; τ) and the case withξ > φ (r; ρτ), where lender strategies in the opt-out segment respectively follow Case 1 and Case 3in Proposition 5; then we characterize the equilibrium in the case with φ (r; τ) ≤ ξ ≤ φ (r; ρτ).Small ξ Case: ξ < φ (r; τ)

Proof. First we show that when τ− ≤ τ , low-type has a higher willingness to sign up. The ξthreshold of lender strategy in the opt-out pool is φ (r; τ−). Note that Condition 29 ensures rτ− ≥1 − xf . When τ− ≤ τ , we have φ (r; τ−) ≥ φ (r; τ) > ξ, so for the opting out borrowers, lenderstrategy and borrower surplus follow Case 1 in Proposition 5. Then for high-type to be willing tosign up,

V ξh (τ−) = Vh (τ−)

willing to sign up≤ (1− ξ)Vh(τ+) = V ξ,OB

h (τ+) .

HenceVh (τ−) < Vh(τ+) and τ− < τ+. With the better inference and the effects of ξ-event on lowtype shown in (23) and (28), low-type must strictly prefer to sign up:

V ξl (τ−) < Vl (τ−) < Vl (τ+) < V ξ,OB

l (τ+).

This result rules out equilibrium where a higher proportion of high-type borrowers sign up, σh ≥σl > 0, under which τ− ≤ τ follows and low-type has higher willingness to sign up. If 1 > σh > 0,then low-type must strictly prefer signing up and σl = 1 > σh. If σh = 1, then σl = 1, butτ− = τ+ = τ contradicts with high type’s sign up incentive.

We now rule out that a larger proportion of low-type signing up in equilibrium, i.e., σl > σh > 0and hence τ− > τ > τ+. In this case, the endogenous ξ threshold φ (r; τ−) < φ (r; τ) and lendercompetition in the opt-out pool may not always follow one case in Proposition 5. If ξ < φ (r; τ−),the competition follows Case 1 in Proposition 5, but τ− > τ > τ+ violates high-type’s sign upincentive. If ξ = φ (r; τ−) we show later that it must be σl < σh; and if φ (r; τ−) < ξ < φ (r; τ), we

44

show later that σh = 1, σl = 0. The last two cases have σl < σh hence contradict with the premisethat “larger proportion of low-type signing up in equilibrium.”

Hence the only possible equilibria is σh = σl = 0 and τ− = τ+ = τ , under which lender strategyis Case 1 in Proposition 5. Introduce τ as the threshold τ+ for high-type to be indifferent to signup, so

Vh (τ) = (1− ξ)Vh(τ).

If the off-equilibrium belief for anyone who signs up satisfy τ+ < τ , high-type borrower does notsign up, and low-type also does not want to sign up to be revealed.

Therefore, in the unique equilibrium nobody signs up and the off-equilibrium belief satisfiesτ+ < τ .

Large ξ Case: ξ > φ (r; ρτ)

Proof. Note that τ− = ρτ is the lower bound of τ−, and is reached when all tech savvy high-typesign up,σh = 1, but none of the low-type signs up, σl = 0. Hence, for any possible equilibriumbelief τ−, we have ξ > φ (r; ρτ) ≥ φ (r; τ−): lender competition for borrowers who did not sign upalways follows Case 3 in Proposition 5.

Eq. (38) says it is a dominant strategy for the l-type borrower not to sign up, σl = 0. Thenif anyone were to sign up, it must be a high-type borrower and τ+ = ∞. As a result, in the non–ξ-event, lenders compete for the opt-in segment a la Bertrand: lenders always charge r = rξ,OB =1−xfτ+

= 0. Then the expected interest rate after open banking is ξr, and is smaller than that beforeopen banking, (1− ξ)E

[min

{rξb , r

ξf

}]+ ξE

[rξf

]:

ξr −[(1− ξ)E

[min

{rξb , r

ξf

}]+ ξE

[rξf

]]= ξr −

[(2− ξ) ξr + (1− ξ)2 (1− xb) (1− xf )

τ−

]= −ξ (1− ξ) r − (1− ξ)2 (1− xb) (1− xf )

τ−< 0.

Therefore in the unique sign-up equilibrium, σh = 1 and σl = 0.

Intermediate ξ Case: φ (r; τ) < ξ < φ (r; ρτ)

Proof. Step 1. We argue that in equilibrium ξ = φ (r; τ−) always holds so that the lender com-petition in the opt-out segment switches structures. Otherwise, if in equilibrium ξ < φ (r; τ−),nobody signs up and τ− = τ , which contradicts with φ (r; τ) ≤ ξ; if ξ > φ (r; τ−), only tech-savvyhigh-type borrowers opt in and τ− = ρτ which contradicts with ξ ≤ φ− (r; τ− = ρτ). Hence, whenφ (r; τ) ≤ ξ ≤ φ (r; ρτ), in equilibrium ξ is on the cutoff ξ = φ (r; τ−).

Step 2. We argue that in equilibrium it must be that σl ∈ (0, 1) and σh > 0. Suppose not; weprove by contradiction.

1. Say σl = 0. If σh = 0, then τ− = τ+ = τ and ξ > φ (r; τ−), lenders compete for the opt-outsegment following Case 3 in Proposition 5, which leads to σh = 1, σl = 0, contradiction. If

45

σh > 0, then τ+ = +∞ and for a borrower who signs up lenders always make an offer uponH; it follows that low-type borrowers must be at least indifferent to sign up, contradiction.

2. Hence, σl > 0 in equilibrium, which implies that some high-type borrowers must sign up (i.e.,σh > 0); otherwise the low-type fully reveal themselves in the opt-in segment and lenders donot participate.

3. We now rule out the case of σl = 1, under which τ− ≥ τ and ξ > φ (r; τ) ≥ φ (r; τ−). Lendercompetition in the opt-out segment leads to sign-up strategies σh = 1, σl = 0, contradiction.

Step 3. Now we derive the equilibrium sign-up behaviors. From φ (r; τ−) = ξ, we have

τ− = 1− xfr

(xbξ

+ 1− xb). (39)

The fintech’s offering probability mξ,OBf− in the opt-out segment and beliefs τ+, τ− make low-type

borrowers indifferent (i.e., 1 > σl > 0) and high-types either indifferent or strictly prefer to sign up(i.e., σh > 0). Specifically, borrower surplus for not signing up are

V ξ,OBh,− (τ−) = Vh (τ−) ,

V ξ,OBl,− (τ−) = (1− ξ)

[1− xb

(xf + (1− xf )

(1−mξ,OB

f−

))]︸ ︷︷ ︸

prob at least one loan

+ξ (1− xf )mξ,OBf−︸ ︷︷ ︸

prob fintech loan

,

where mξ,OBf− versus mass point at r = r only influences the probability of receiving a loan but

does not affect the expected interest rate. For borrowers who signed up, surplus V ξ,OBi,+ (τ+) are the

same as (27) and (28) except for adjusted belief τ+.For the high-type, there are two subcases to consider.

1. Suppose that the high-type are indifferent to sign up; then τ+ and mξ,OBf− ≥ 1−φ (r; τ−) make

both type of borrowers indifferent:35 V ξ,OBi,− (τ−) = V ξ,OB

i,+ (τ+) , i = h, l. Hence, we have36

φ+ (r) =2xb + (1 + ξ) (1− xb)−

√[2xb + (1 + ξ) (1− xb)]2 − 4ξ (xb + (1− xb) ξ)2 (xb + (1− xb) ξ)

, (41)

35Equilibrium mξ,OBf− is well defined and unique. The low-type’s indifference condition is equivalent to

∆V ξ,OBl = (1− xf )[(

1−mξ,OBf−

)(ξ + (1− ξ)xb)− (1− ξ)xbφ+ (r)

],

where φ+ (r) ≡ φ (r; τ+) .mξ,OBf− ≥ 1 − φ (r; τ−) is satisfied because when mξ,OB

f− = 1 − φ (r; τ−) low type strictlyprefers to sign up as φ− (r) (ξ + (1− ξ)xb) > (1− ξ)xb φ+ (r)︸ ︷︷ ︸

<φ−=ξ

, and when mξ,OBf− = 1, low type strictly prefers to opt

out. Note that ∆V ξ,OBl is monotone in mξ,OBf− , so mξ,OB

f− is unique.36It follows that φ+ (r) satisfies the following quadratic equation,

(xb + (1− xb) ξ)φ2+ − [2xb + (1 + ξ) (1− xb)]φ+ + ξ = 0. (40)

It has two positive roots, and only the smaller root is smaller than 1. Later we study τ+; since τ+ and φ+are negativelyrelated, τ+ takes the larger root.

46

andmξ,OBf− = 1− (1− ξ)xbφ+ (r)

ξ + (1− ξ)xb. (42)

From belief updating rules τ+ = τ σhσl and τ− = τ ρ+(1−ρ)(1−σh)ρ+(1−ρ)(1−σl) ,we solve for

σh =τ+τ−− τ+

τ

(1− ρ)(τ+τ−− 1

) and σl =ττ−− 1

(1− ρ)(τ+τ−− 1

) , (43)

where τ− is determined in (39) and τ+ is determined by (41) and (6). Note that ∂∂τ−

φ (r; τ−) <0 implies that τ− < τ < τ+ for ξ > φ (r; τ) and φ (r; τ−) = ξ. As a result, σh > σl from beliefupdating rule τ+ = τ σhσl . This observation completes the earlier proof of a unique sign-upequilibrium under “ Small ξ Case” (i.e., ξ < φ (r; τ)) where we rule out τ+ > τ > τ− withσl > σh.

2. Now suppose that σh = 1. From belief updating we have

σl =1− τ

τ−ρ

1− ρ , and τ+ = τ (1− ρ)1− τ

τ−ρ,

and mξ,OBf− is determined in (42). Note that this corner equilibrium must arise when ξ →

φ (r; ρτ); in this situation, we have τ− → ρτ and τ+ → +∞, under which

V ξ,OBh,+ (τ+)→ (1− ξ) r > V ξ,OB

h,− (τ−) = (1− φ (r; τ−))︸ ︷︷ ︸=1−ξ

(r − r−) ,

and the high-type borrowers strictly prefer to sign up. At the same time, σl → 0 whilelow-type borrowers stay indifferent whether or not to sign up.

A.12 Proof of Proposition 8

Proof. First we argue that there exists a ξ ∈ (φ (r; τ) , φ (r; ρτ)) such that 0 ≤ σh < 1 whenφ (r; τ) ≤ ξ < ξ and σh = 1 when ξ ≤ ξ ≤ φ (r; ρτ). To see this, we already argued in AppendixA.11 that σh = 1, 0 < σl < 1 must arise when ξ is sufficiently close to φ (r; ρτ). When ξ = φ (r; τ),we have τ− = τ and σh = σl = 0. Hence by continuity of σh, σl in ξ, there exists such a ξ belowwhich high-type is indifferent to sign up and above which high-type strictly prefers to sign up.

Recall that θ+, θ− are respectively the average quality of opt-in and opt-out borrowers. Natu-rally p (ξ) θ+ + (1− p (ξ)) θ− = θ, and thus

p (ξ) = θ − θ−θ+ − θ−

47

decreases in both θ+ and θ−. When φ (r; τ) ≤ ξ < ξ, high-type is indifferent to sign up, so τ+

must decrease with ξ to balance the deterioration of τ−. Hence, p (ξ) increases in ξ in this case.On the other hand, when ξ ≥ ξ, we have σh = 1 and σl =

1− ττ−ρ

1−ρ , so p (ξ) ≡ (1− ρ) [θ + (1− θ)σl]decreases in ξ.

Then we discuss the welfare implications for ξ ∈[φ (r; τ) , ξ

]. First, since tech savvy borrowers of

both credit type are indifferent to sign up and have the same surplus as non--tech-savvy borrowers.Hence, it suffices to discuss how open banking affects the non--tech-savvy borrowers. We arguethat high-type loses,

∆V ξ,OBh = Vh (τ−)︸ ︷︷ ︸

V ξ,OBh,ρ

− (1− ξ)2[r − (1− xb) (1− xf )

τ

]︸ ︷︷ ︸

V ξh,ρ

(·)

= (1− ξ)2

τ

(1− xb) (1− xf )xbxb + ξ (1− xb)

[1− ξ

φ(r; τ)

]< 0.

Note that before open banking lenders always make loans upon H signal when ξ ≥ φ (r; τ), solow-type borrowers are hurt by open banking: ∆V ξ,OB

l < 0. Therefore, all borrowers are hurt byopen banking when ξ ∈

[φ (r; τ) , ξ

]even if they voluntarily choose whether or not to sign up.

Now we study firm profits. In the region of φ (r; τ) ≤ ξ ≤ φ (r; ρτ), we show that the openbanking hurts the bank while benefits the fintech. To see this, the profits of two lenders after openbanking,

πξ,OBb = πξ,OBb+ + πξ,OBb− = n+ (1− ξ) xb − xf1 + τ+︸ ︷︷ ︸=πξ,OB

b+

+n− (1− ξ) xb − xf1 + τ−︸ ︷︷ ︸=πξ,OB

f−

= (1− ξ) xb − xf1 + τ,

πξ,OBf = πξ,OBf+ + πξ,OBf−︸ ︷︷ ︸=0

= ξ [θ (1− ρ)σhr − (1− θ) (1− ρ)σl (1− xf )] ;

while their profits before open banking are

πξb = (1− ξ)[ξrτ

1 + τ− ξ 1− xb

1 + τ− (1− ξ) (1− xb)xf

1 + τ

],

πξf = ξθr − ξ (1− θ) (1− xf )− (1− ξ) (1− θ)xb(1− xf ).

We hence have that

∆πξ,OBb ≡ πξ,OBb − πξb = (1− ξ) (1− xf )xb1 + τ

(1− ξ

φ (r; τ)

)< 0;

∆πξ,OBf = πξ,OBf − πξf = ξ (1− xf )1 + τ

(1− ρ)σl

(rτ−

1− xf− 1

)> 0.

Finally we study the total profits for the financial sector ∆πξ,OBb + ∆πξ,OBf , and give sufficientconditions for it to rise after open banking. Note that φ (r; τ−) = ξ implies that rτ−

1−xf − 1 = ξxb1−ξ ,

48

we have

∆πξ,OBb + ∆πξ,OBf = (1− ξ) (1− xf )xb1 + τ

[1− φ (r; τ−)

φ (r; τ) + (1− ρ)σl].

From (43), we have (1− ρ)σl = τ−τ−τ+−τ− , and hence the

1− φ (r; τ−)φ (r; τ) + (1− ρ)σl = (τ − τ−)

(τ+ − τ−) (rτ− − (1− xb) (1− xf ))︸ ︷︷ ︸positive

(2rτ− − rτ+ − (1− xb) (1− xf ))

(44)

Hence it boils down to the sign of the last bracket in Eq. (44). Notice that while τ− = τ is continuousat ξ = φ (r; τ), τ+ = τ σhσl typically jumps upward at ξ = φ (r; τ) from left. So it is non-trivial toshow that the total financial sector gains even when in the neighborhood of ξ = φ (r; τ).

We use the high-type’s indifference curve, which says

1− 1− xfrτ−

=(

1− 1− xfrτ+

)(rτ+ − (1− xf )

rτ+ − (1− xf ) (1− xb)

)⇔Q (τ+) ≡ (rτ+)2 − [(1 + xb) rτ− + (1− xb) (1− xf )] rτ+ + (1− xf ) rτ− = 0.

We then try to ensure that τ+ < 2τ−−(1−xb)(1−xf)

r (so that the last bracket in Eq. (44) is positive),

by checking the sign of Q(τ+ = 2τ− −

(1−xb)(1−xf)r

), which equals

(2rτ− − (1− xb) (1− xf ))2 − [(1 + xb) rτ− + (1− xb) (1− xf )] (2rτ− − (1− xb) (1− xf )) + (1− xf ) rτ−

=rτ−

2 (1− xb) rτ− − (5− xb) (1− xb) (1− xf ) + (1− xf )︸ ︷︷ ︸M(ξ)

+ 2 (1− xb)2 (1− xf )2︸ ︷︷ ︸>0

.

Because Q (·) is quadratic and open-upward, and we take the larger solution (see footnote 36), toensure Q

(τ+ = 2τ− −

(1−xb)(1−xf)r

)> 0 we need

M (ξ) ≡ 2 (1− xb) rτ− (ξ)− (5− xb) (1− xb) (1− xf ) + (1− xf ) > 0

for τ− (ξ) when ξ ∈[φ (r; τ) , ξ

]. (Note, (1− xb)2 (1− xf )2 will be at higher order when xj ’s are

close to 1, hence can be ignored). Because τ− is decreasing in ξ, it is equivalent to ensure thatM (·) > 0 at both ends.

1. When ξ = φ (r; τ), τ− (ξ) = τ , so we require that (recall ρrτ ≥ 1− xf in (29))

2ρ

(1− xb)− (5− xb) (1− xb) + 1 > 0. (45)

49

2. When ξ = ξ, we have σh = 1 which implies that

τ−(ξ)

=τ+(ξ)

τ+(ξ)− (1− ρ) τ

ρrτ ≥τ+(ξ)

τ+(ξ)− (1− ρ) τ

(1− xf ) .

So M(ξ)> 0 requires that

2 (1− xb)τ+(ξ)

τ+(ξ)− (1− ρ) τ

− (5− xb) (1− xb) + 1 > 0, (46)

which is easy to verify ex post once we solved for τ+(ξ).

In sum, the simple conditions (45) and (46) guarantee that the financial sector gains after openbanking when ξ ∈

[φ (r; τ) , ξ

](note, these conditions are also necessary if ρrτ = 1 − xf and for

sufficiently large xj ’s).

50

B Online Appendix

B.1 Properties of Mixed-Strategy Equilibria

Here we show that in our baseline model any mixed-strategy equilibrium is well behaved.

Lemma 3. In any mixed-strategy equilibrium, the two lenders’ interest rate distributions have thefollowing properties:

1. they share the same lower bound r > 0 and the same upper bound r in their supports;

2. they have no gaps in their supports;

3. they have no mass points except that one of them can have one at r.

Proof. Here we show the properties of a mixed-strategy equilibrium. (i) For the lower bound result,suppose in contrast that, say, rw < rs, i.e. Fw has a smaller lower bound than Fs. Then for theweak lender, offering r ∈ (rw, rs) is always more profitable than offering rw since both lead to thesame demand. This contradicts rw being in the support. For the upper bound result, supposefirst max{rw, rs} < r. If rw 6= rs, then for the lender with the higher upper bound, offering aninterest rate slightly above its upper bound will be a profitable deviation. If rw = rs, since it isimpossible that both distributions have a mass point at this upper bound, at least one lender willhave an incentive to slightly raise its interest rate without losing any demand. Now suppose, say,rw < r and rs = r. Then the support of Fs should have a gap in (rw, r) since any interest rate inthis interval is dominated by r. But this is impossible in equilibrium since for at least one lenderoffering an interest rate slightly above rw would be a profitable deviation. This proves rw = rs = r.

(ii) Suppose that, say, the support of Fs has a gap (r1, r2) ⊂ [r, r]. Then Fw should have noweight in this interval either as any r ∈ (r1, r2) will lead to the same demand for the weak lenderand so a higher r will be more profitable. If neither distribution has a mass point at r1, unilaterallyoffering r ∈ (r1, r2) will be a profitable deviation for either lender. If one distribution, say, Fs hasa mass point at r1, unilaterally offering r ∈ (r1, r2) will be a profitable deviation for the stronglender. It is impossible that both distributions have a mass point at r1.

(iii) Suppose first that one distribution, say, Fs has a mass point at r = r. Then it will be aprofitable deviation for the weak lender to offer an interest rate slightly below r. Suppose then thatFs has a mass point at r ∈ (r, r). Then for the weak lender an interest rate just slightly below r

should be more profitable than any interest rate in [r, r+ ε] for some 0 < ε < r− r. In other words,the support of Fw must have a gap in this interval. This, however, is impossible as we have shownin (ii). Finally it is impossible that both distributions have a mass point at r.

51

B.2 Proof of Proposition 4

As this part of proof is relatively independent, for notational convenience we use α ≡ xb to denotethe traditional bank’s screening ability, and β ≡ xf , γ ≡ x′f respectively for the fintech’s screeningabilities before and after open banking.

There are three sets primitive parameters that are relatively independent in the conditions thatwe need. The first set is

Γ ≡ (α, β, γ, θ) ;

the second is r, which mainly affects lender profits; and the last is the proportion of privacyconcerned borrowers ρ, which affects the posterior credit quality τ−, τ+ for given borrower decisionsσh, σl.

We focus on equilibrium where for borrowers without privacy concern, h type always sign upfor open banking and l type is indifferent whether or not to sign up. For illustration purpose, werewrite borrower surplus function as (note σl will be endogenously determined in equilibrium)

Vi (Γ, r, ρ;σl) ≡ Vi

where i = h or l. Specifically, before open banking,

Vi (Γ, r) ≡ Vi (α, β, θ) ;

after open banking, for the group of borrowers who signed up, we denote

V +i (Γ, r;σl) ≡ Vi

(α, γ, τ+ = τ

σl

),

for the borrowers who did not sign up, we denote

V −i (Γ, r, ρ;σl) ≡ Vi(α, β, τ− = ρτ

1− σl + ρσl

).

Note that the private benefit of receiving the loan does not affect borrowers’ sign up decisions.For notational convenience, we assume vh = 0 and vl = 1 when discussing borrowers’ sign up.

B.2.1 Proof of Proposition 4 Part 1 (Existence)

Step 1.We find the parameter conditions under which there exists an equilibrium with σl ∈ (0, 1) where

1) for borrowers without privacy concerns, l-type is indifferent whether or not to sign up, h-typestrictly prefers signing up, and 2) lenders are willing to participate. We show that 1) implies anone-to mapping between σl and ρ, while 2) implies that σl and ρ have to satisfy another condition.

Because of the l’s indifference 1), in later proofs we can treat σl ∈ (0, 1) as exogenous, derive

52

its range that satisfies the desired property, and later transform this range to that for the primitiveparameter ρ.

Step 1.1Define

∆Vl (σl, ρ) ≡ V +l (Γ, r;σl)− V −l (Γ, r, ρ;σl) ,

which satisfies

∂∆Vl (σl, ρ)∂σl

= ∂Vl (α, γ, τ+ (σl))∂τ+︸ ︷︷ ︸

+

∂τ+∂σl︸ ︷︷ ︸−

− ∂Vl (α, γ, τ− (σl; ρ))∂τ−︸ ︷︷ ︸

+

∂τ−∂σl︸ ︷︷ ︸+

< 0, (47)

∂∆Vl (σl, ρ)∂ρ

= − ∂Vl (α, γ, τ− (σl; ρ))∂τ−︸ ︷︷ ︸

+

∂τ−∂ρ︸ ︷︷ ︸+

< 0. (48)

Hence, if for some ρ we have

∆Vl (σl = 0, ρ) > 0, (49)

∆Vl (σl → 1, ρ) < 0, (50)

then there must exist a σl ∈ (0, 1) such that

∆Vl (σl = σl, ρ) = 0, (51)

and this defines a function ρ = ρ (σl) such that ∆Vl (σl, ρ (σl)) = 0. From implicit function theoremwe know ρ (σl) strictly decreases in σl:

∂ρ (σl)∂σl

= −∂∆Vl

(σl,ρ)∂σl

∂∆Vl(σl,ρ)∂ρ

< 0.

And, when γ > α, from Lemma 2 we know that high type must strictly prefer signing up.Step 1.2We now ensure lenders’ lender participation constraints, both before open banking and after

open banking (regardless of facing borrowers who signed up or didn’t). If

rτ− = rτ1−σlρ + σl

≥ 1− β, (52)

lender participation is satisfied for borrowers who did not sign up. Because τ− ≤ τ ≤ τ+ andβ < α, lenders are also willing to participate upon H in the sign-up borrower pool and before openbanking.

For any (σl, ρ) that satisfy (51) (or ρ = ρ (σl)), because V −l strictly increases in rτ−, (52) is

53

equivalent toV +l (Γ, r; σl) = V −l (Γ, r, ρ; σl) ≥ 1− α. (53)

The advantage of (53) is to derive a condition of σl to be satisfied that is free of ρ, which combinedwith ρ = ρ (σl) imply (52). As V +

l strictly decreases in σl, (53) is equivalent toσl ≤ ¯lσ (Γ, r) and

ρ = ρ (σl), where ¯lσ (Γ, r) is defined as

V +l

(Γ, r; ¯

lσ (Γ, r))

= 1− α. (54)

Then the resulting equilibrium satisfies σh = 1 and

σl ∈(0,min

(¯lσ (Γ, r) , 1

)). (55)

In sum, (55) translates to a non-empty set of ρ = ρ (σl), (σl > 0 says ρ < ρ (Γ, r) ≡ ρ (0), which isequivalent to (49)) together with (50) that we will ensure later, we have shown the existence of thedesired equilibrium.

Step 2.We now impose conditions under which high-type to be hurt by open banking, i.e.,

Vh (Γ, r) > V +h (Γ, r; σl) . (56)

This implies that everyone is hurt by open banking: anyone who does not sign up is perceived tobe of a lower average quality than the whole population, i.e., τ− < θ, and thus worse off as ∂Vi

∂θ > 0;l-type who signed up receive the same payoff as when they do not sign up, and thus also hurt byopen banking. As ∂V +

h(Γ,r;σl)∂σl

< 0, (56) requires that

σl > z (Γ, r)

where z (Γ, r) is defined as Vh (Γ, r) = V +h (Γ, r; σl = z (Γ, r)). We know that z > 0.37 From (55),

we need to make sure that z < min{

¯lσ (Γ, r) , 1

}. The condition for z < 1 is

Vh (Γ, r) > V +h (Γ, r; σl = 1) ; (57)

and the condition for z < ¯lσ (Γ, r) (defined in (54)), is

Vh (Γ, r) > V +h

(Γ, r; σl = ¯

lσ (Γ, r)). (58)

Step 2.1One can show that (57) implies our previous condition (50). (57) says(

r − 1− βτ

)(1− φ (r;α, β, θ)) >

(r − 1− α

τ

)(1− φ (r;α, γ, θ)) ,

37This is because Vh (Γ, r) < r − 1−βτ

< V +h (Γ, r; σl = 0) = r − 1−α

τ.

54

and since 1−βτ > 1−α

τ , we have φ (r;α, β, θ) < φ (r;α, γ, θ). As a result (recall φ ∈ (0, 1) andβ < α < γ, and third inequality uses φ (r;α, β, θ) < φ (r;α, γ, θ))

V +l (Γ, r;σl = 1) = 1− γ [α+ (1− α) · φ (r;α, γ, θ)]

< 1− α [α+ (1− α) · φ (r;α, γ, θ)]

< 1− α [β + (1− β) · φ (r;α, γ, θ)]

< 1− α [β + (1− β) · φ (r;α, β, θ)]

= V −l (Γ, r, ρ;σl = 1) .

Step 2.2We now show that (57) and (58) correspond to a non-empty set of {Γ, r}. Let R (r) ≡ rτ

1−β −1 >0, which can be viewed as a function of r given other parameters. (57) is equivalent to

G1 (R;α, β, γ) ≡(γ

1− α1− β −

α− β1− β − α

)R2 −

(α− β1− β

)[(α− β1− β

)+ 2α

]R− α

(α− β1− β

)2> 0,(59)

and (58) is equivalent to

G2 (R;α, γ) ≡ R2 − (γ − α)2 (α+ 1)α (1− γ) (1 + γ − 2α) ·R−

(γ − α)2

(1− γ) (1 + γ − 2α) > 0. (60)

38

Our goal is to give a condition on R so that both (59) and (60) hold. A necessary condition for(59) is that

β > β1 (α, γ) ≡ 2α− γ (1− α)1 + α

,

under which we have γ 1−α1−β −

α−β1−β − α > 0. Then, because both G1 (R) = 0 and G2 (R) = 0 are

quadratic functions with one negative root and one positive root, the condition that we are after is

R (r) > max {R1 (α, β, γ) , R2 (α, γ)} , (61)

where R1 (α, β, γ) is the positive root of G1 (R;α, β, γ) = 0 and R2 (α, γ) is the positive root ofG2 (R;α, γ) = 0.

38Plug (54) into (58) we have R2

(R+1)(R+α) >

[1− 1

γ2(1−α)α(1−γ) +1−γ

] [1− α(1−γ)

γ(1−α)

]. Because

0 <

1− 1

γ2(1− α)α(1− γ)︸ ︷︷ ︸

>γ

+1− γ

1− α(1− γ)

γ(1− α)︸ ︷︷ ︸<1

< 1,

this corresponds to an inequality that says an upward quadratic function of R is positive.

55

Step 3.To sum up, if {Γ, ρ, r} satisfy conditions i) γ > α > β > β1 (α, γ) ≡ 2α−γ(1−α)

1+α , ii) (57) and (58)for r, which is summarized by (61), and finally iii) σl ∈

(z (Γ, r) ,min

(¯lσ (Γ, r) , 1

))that imply

a compact set of ρ = ρ (σl), then there exists an equilibrium such that σh = 1, σl ∈ (0, 1), andeveryone is hurt by open bankingB.2.2 Proof of Proposition 4 Part 1 (Uniqueness)

Generally there are nine possible types of equilibrium under the binary sign-up cost specification:

[σh = 1, σl = 1] [σh = 1, σl ∈ (0, 1)] [σh = 1, σl = 0][σh ∈ (0, 1) , σl = 1] [σh ∈ (0, 1) , σl ∈ (0, 1)] [σh ∈ (0, 1) , σl = 0][σh = 0, σl = 1] [σh = 0, σl ∈ (0, 1)] [σh = 0, σl = 0]

where the equilibrium that we construct is boldfaced. Notice that [σh = 0, σl = 0] could be sup-ported by any consistent off-equilibrium belief of θ+. In this part, we argue that if (62) holdsadditionally, then except for the special equilibrium [σh = 0, σl = 0], the equilibrium constructedin Part 1 is unique.

The additional condition we need for uniqueness is

R (r) < R3 (α, β, γ) ≡ α( 1− γγ − β

), (62)

which is equivalent to

Vl(α, γ, θ+ = 1) > Vl(α, β, θ) (63)

once we plug in borrower surplus (9) and (10). Note that even when θ+ = 1, we assume that firmsstill screen the consumer who signs up. As there is no cost in screening, this assumption leads toa more general argument, whereas (63) is automatically satisfied if firms do not screen and alwaysserve the sign-up consumer given θ+ = 1. (63) says l (0) will sign on if he is perceived to be h-typewhen doing so, relative to being perceived as the average type θ by not signing on.

We need to verify that the set of primitive parameters that satisfies all the sufficient conditionsis non-empty. In light of (61), we need that

R3 (α, β, γ) > max {R1 (α, β, γ) , R2 (α, γ)} , (64)

and we aim to provide a sufficient condition under which (64) holds.For any γ > α, if β = α, we have R1 (α, β = α, γ) = 0 < R2 (α, γ). Recall that for R >

R1 (α, β, γ) to be sufficient for (59), we need β > β1 (α, γ) ≡ 2α−γ(1−α)1+α with β1 (α, γ) < α. Note

that we have R1(α, β → β1 (α, γ)+ , γ

)= +∞ > R2 (α, γ). Define

β (α, γ) ≡ sup {β : R1 (α, β, γ) = R2 (α, γ) , β1 (α, γ) < β < α} , (65)

56

which must satisfy β1 (α, γ) < β (α, γ) < α strictly. Then when β ∈(β (α, γ) , α

), we have

R2 (α, γ) > R1 (α, β, γ), and therefore (64) is equivalent to

R3 (α, β, γ) > R2 (α, γ) . (66)

Note that ∂R3(α,β,γ)∂β > 0; then a sufficient condition for (66) where β > β (α, γ) > β1 (α, γ) is

R3 (α, β1 (α, γ) , γ) > R2 (α, γ) . (67)

The condition (67) only involves α and γ, and we derive the restriction on the space of {α, γ} sothat (67) holds.

Because R3 (α, β1 (α, γ) , γ) = α(1−γ)(1+α)2(γ−α) > 0, and G2 (R) = 0 is a quadratic function with one

negative root and one positive root, (67) is equivalent to

G2 (R3 (α, β1 (α, γ) , γ)) = α2 (1− γ)2 (1 + α)2

4 (γ − α)2 − (γ − α) (α+ 1)2

2 (1 + γ − 2α) −(γ − α)2

(1− γ) (1 + γ − 2α)

≡ G2 (γ;α) > 0.

One can easily show ∂G2(γ;α)∂γ < 0.39 Together with

limγ→α−

G2 (R3 (α, β1 (α, γ) , γ))→ +∞,

we know there exists a γ (α) > α such that when α < γ < γ (α), (67) is satisfied.

39Let g1 (γ;α) ≡ 1−γγ−α , and g2 (γ;α) = γ−α

1+γ−2α ; then with 1 > γ > α > 0 we have

∂g1 (γ;α)∂γ

< 0, ∂g2 (γ;α)∂γ

> 0.

As a result, the following function is strictly decreases in γ:

G2 (γ;α) = α2 (1 + α)2

4 · g1 (γ;α)2 − (1 + α)2

2 · g2 (γ;α)− g2 (γ;α)g1 (γ;α) .

57